This answer is based upon united-states law. Outside the United States that law may, and indeed, is likely to, differ, as the legal analysis in U.S. law is unusual in multiple respects with regard to these issue.
The premise of the question is basically incorrect. There is not a stark legal definitional distinction between physical abuse and psychological abuse.
Child abuse and neglect are defined by Federal and State laws. At the
State level, child abuse and neglect may be defined in both civil and
criminal statutes. This publication presents civil definitions that
determine the grounds for intervention by State child protective
agencies.
At the Federal level, the Child Abuse Prevention and Treatment Act
(CAPTA) has defined child abuse and neglect as "any recent act or
failure to act on the part of a parent or caregiver that results in
death, serious physical or emotional harm, sexual abuse, or
exploitation, or an act or failure to act that presents an imminent
risk of serious harm. CAPTA Reauthorization Act of 2010 (P.L.
111-320), 42 U.S.C. § 5101, Note (§ 3).
(Source).
This definition and other definitions were contained in the 2010 amendments to the Act, but were not codified in the United States Code's text.
State definitions very considerably, but significantly overlap with the CAPTA definition. For example, a non-exclusive list of conduct that constitutes misdemeanor or low level felony child abuse in Colorado if you engage in it includes:
you are in a position of trust in relation to the child, and
you participate in a continued pattern of conduct that results in the child’s malnourishment;
you fail to ensure the child’s access to proper medical care;
you participate in a continued pattern of cruel punishment or unreasonable isolation or confinement of the child;
you make repeated threats of harm or death to the child or to a significant person in the child’s life in the presence of the child;
you commit a continued pattern of acts of domestic violence in the presence of the child; or
you participate in a continued pattern of extreme deprivation of hygienic or sanitary conditions in the child’s daily living
environment.
(The criminal child abuse statute, Colorado Revised Statutes § 18-6-401, is somewhat tricky to parse).
Some of this conduct causes emotional harm more than physical harm.
As is typical, this only moderately overlaps with the civil standard for termination of parental rights in Colorado pursuant to Colorado Revised Statutes §§ 19-3-102 and 19-5-105 which state in the pertinent parts:
C.R.S. § 19-3-102:
(1) A child is neglected or dependent if:
(a) A parent, guardian, or legal custodian has abandoned the child or
has subjected him or her to mistreatment or abuse or a parent,
guardian, or legal custodian has suffered or allowed another to
mistreat or abuse the child without taking lawful means to stop such
mistreatment or abuse and prevent it from recurring;
(b) The child lacks proper parental care through the actions or
omissions of the parent, guardian, or legal custodian;
(c) The child's environment is injurious to his or her welfare;
(d) A parent, guardian, or legal custodian fails or refuses to provide
the child with proper or necessary subsistence, education, medical
care, or any other care necessary for his or her health, guidance, or
well-being;
(e) The child is homeless, without proper care, or not domiciled with
his or her parent, guardian, or legal custodian through no fault of
such parent, guardian, or legal custodian;
(f) The child has run away from home or is otherwise beyond the
control of his or her parent, guardian, or legal custodian;
(g) The child tests positive at birth for either a schedule I
controlled substance, as defined in section 18-18-203, C.R.S., or a
schedule II controlled substance, as defined in section 18-18-204,
C.R.S., unless the child tests positive for a schedule II controlled
substance as a result of the mother's lawful intake of such substance
as prescribed.
(2) A child is neglected or dependent if:
(a) A parent, guardian, or legal custodian has subjected another child
or children to an identifiable pattern of habitual abuse; and
(b) Such parent, guardian, or legal custodian has been the respondent
in another proceeding under this article in which a court has
adjudicated another child to be neglected or dependent based upon
allegations of sexual or physical abuse, or a court of competent
jurisdiction has determined that such parent's, guardian's, or legal
custodian's abuse or neglect has caused the death of another child;
and
(c) The pattern of habitual abuse described in paragraph (a) of this
subsection (2) and the type of abuse described in the allegations
specified in paragraph (b) of this subsection (2) pose a current
threat to the child.
C.R.S. § 19-5-105:
(3) If, after the inquiry, the other birth parent is identified to the
satisfaction of the court or if more than one person is identified as
a possible parent, each shall be given notice of the proceeding in
accordance with subsection (5) of this section, including notice of
the person's right to waive his or her right to appear and contest. If
any of them waives his or her right to appear and contest or fails to
appear or, if appearing, cannot personally assume legal and physical
custody, taking into account the child's age, needs, and individual
circumstances, such person's parent-child legal relationship with
reference to the child shall be terminated. If the other birth parent
or a person representing himself or herself to be the other birth
parent appears and demonstrates the desire and ability to personally
assume legal and physical custody of the child, taking into account
the child's age, needs, and individual circumstances, the court
shall proceed to determine parentage under article 4 of this title. If
the court determines that the person is the other birth parent, the
court shall set a hearing, as expeditiously as possible, to determine
whether the interests of the child or of the community require that
the other parent's rights be terminated or, if they are not
terminated, to determine whether:
(a) To award custody to the other birth parent or to the physical
custodian of the child; or
(b) To direct that a dependency and neglect action be filed pursuant
to part 5 of article 3 of this title with appropriate orders for the
protection of the child during the pendency of the action.
(3.1) The court may order the termination of the other birth parent's
parental rights upon a finding that termination is in the best
interests of the child and that there is clear and convincing evidence
of one or more of the following:
(a) That the parent is unfit. In considering the fitness of the
child's parent, the court shall consider, but shall not be limited to,
the following:
(I) Emotional illness, mental illness, or mental deficiency of the
parent of such duration or nature as to render the parent unlikely,
within a reasonable period of time, to care for the ongoing physical,
mental, and emotional needs of the child;
(II) A single incident of life-threatening or serious bodily injury or
disfigurement of the child or other children;
(III) Conduct toward the child or other children of a physically or
sexually abusive nature;
(IV) A history of violent behavior that demonstrates that the
individual is unfit to maintain a parent-child relationship with the
minor, which may include an incidence of sexual assault, as defined in
section 19-1-103 (96.5), that resulted in the conception of the child;
(V) Excessive use of intoxicating liquors or use of controlled
substances, as defined in section 18-18-102 (5), C.R.S., that affects
the ability of the individual to care and provide for the child;
(VI) Neglect of the child or other children;
(VII) Injury or death of a sibling or other children due to proven
abuse or neglect by such parent;
(VIII) Whether, on two or more occasions, a child in the physical
custody of the parent has been adjudicated dependent or neglected in a
proceeding under article 3 of this title or comparable proceedings
under the laws of another state or the federal government;
(IX) Whether, on one or more prior occasions, a parent has had his or
her parent-child legal relationship terminated pursuant to this
section or article 3 of this title or comparable proceedings under the
laws of another state or the federal government.
(b) That the parent has not established a substantial, positive
relationship with the child. The court shall consider, but shall not
be limited to, the following in determining whether the parent has
established a substantial, positive relationship with the child:
(I) Whether the parent has maintained regular and meaningful contact
with the child;
(II) Whether the parent has openly lived with the child for at least
one hundred eighty days within the year preceding the filing of the
relinquishment petition or, if the child is less than one year old at
the time of the filing of the relinquishment petition, for at least
one-half of the child's life; and
(III) Whether the parent has openly held out the child as his or her
own child.
The items in bold have, or could sometimes have, a significant emotional well-being component.
In practice, however, this is limited by the constitutional right to raise one's children without undue government interference in circumstances where there is not an imminent risk of serious harm, under the substantive due process doctrine dimensions of the 14th Amendment due process clause,
This is especially true when one's child rearing methods of a religious basis implicating the Free Exercise clause of the 1st Amendment to the U.S. Constitution as incorporated against the states through the 14th Amendment to the U.S. Constitution.
While these constitutional defenses can be asserted in both cases of alleged physical abuse and alleged psychological abuse, these defenses are particularly hard to penetrate in cases of psychological abuse. In particular, In Pierce v. Society of Sisters, 268 U.S. 510 (1925), the "Supreme Court also recognized a substantive due process right 'to control the education of one's children', thus voiding state laws mandating for all students to attend public school." It said:
We think it entirely plain that the Act of 1922 unreasonably
interferes with the liberty of parents and guardians to direct the
upbringing and education of children under their control. As often
heretofore pointed out, rights guaranteed by the Constitution may not
be abridged by legislation which has no reasonable relation to some
purpose within the competency of the state. The fundamental theory of
liberty upon which all governments in this Union repose excludes any
general power of the state to standardize its children by forcing them
to accept instruction from public teachers only.
These are sometimes described as "privacy rights" (and also include the right to legal contraception and the abortion rights of Roe v. Wade), but in this context, a "privacy right" is not the right to keep something unknown to the general public in the literal sense of the words. Instead, it is a privacy right in the less common sense of the words meaning a right to autonomy and freedom of conscience of a parent, associated with the underlying purposes of other constitutional rights that protect more literal forms of privacy.
The other issue is that there is less of a consensus concerning what constitutes psychological abuse sufficiently clearly that it is publicly sanctionable, than there is concerning what constitutes physical abuse. Striking a child for reasons other than to improve a child's behavior is usually considered physical child abuse. Intentionally undermining a child's self-esteem, in contrast, for example, can be justified in myriad ways.