I'm not sure if the information is accurate, but according to the
above text, when a wife cheats, it is marriage not biology that
decides the paternity of the child. I got some questions:
If my wife cheats on me, I would still be the legal parent of the
child. If I don't want this paternity, is there a legal process to
disavow it?
In most states, yes (I can't think of any exceptions, but there are 50 states and more self-governing territories and this is a matter of state law). Typically there is a statute of limitations of one to five years from the date of birth for a husband or person listed as a father on a birth certificate to bring a legal action to disavow paternity. See, e.g., California Family Code §§ 7540-7541 (setting a two year statute of limitations from a child's birth for a person with standing to dispute that a cuckolded husband is the legal father with genetic evidence).
Note also that the process and statute of limitations are usually not the same, if, for example, a child wishes to prove that the child's biological father is someone other than the legally presumed father of that child.
If I fall in love with a married woman and we give birth to a child, I
am the biological but not legal parent of the child. Is there a legal
process for me to claim paternity of the child from the woman's
husband?
Sometimes yes, and sometimes no. The U.S. Supreme Court, in Michael H. v. Gerald D., 491 U.S. 110 (1989), held that a state is not constitutionally required to make such a process available, but some states do anyway.
The details of how this plays out under New York State law are described in this Law.SE question and answer. An analysis of the relevant portions of Minnesota law can be found here.
For example, in California, Family Code Section 7541 limits standing to dispute paternity to spouses, people "presumed to be a parent" under Family Code Section 7611, or representatives of children seeking to establish or disestablish the paternity of someone "presumed to be a parent" under Family Code Section 7611. So, the only people eligible to be found to be parents are (excluding spent provisions of only historical interest):
A husband who was married to the mother at the time of the birth or within 300 days before the birth. § 7611(a).
A putative husband who would have been a spouse under § 7611(a), who marriage is annulled (e.g. because a marriage license expired or a husband was too closely related to the mother or either spouse is already married). § 7611(b).
A putative husband who cohabited with the mother within 300 days before the birth whose attempt to married was too obviously defective to require an annulment (e.g. two fifteen year olds who have a church wedding without a marriage license). § 7611(b).
A husband of the mother who marries the mother after the birth and is also named as a father on the birth certificate, in a voluntary written promise, or in a court order. § 7611(c).
A putative husband who attempt to marry the mother after the birth and is named as a father on the birth certificate, in a voluntary written promise, or in a court order. § 7611(c).
A person "who receives the child into his or her home and openly holds out the child as his or her natural child." § 7611(d).
A parent who dies while the "child is in utero" is this is established in a probate court proceeding. § 7511(f).
Thus, in California, a father of the child of a woman married to someone else, who is still alive, (or someone of behalf of the child seeking to establish that he is the father) can only dispute the paternity of the husband of the child's mother (if the husband himself or the mother does not challenge the husband's paternity) if he "receives the child into his home and openly holds out the child as his or her natural child." Simply claiming paternity without "receiving the child into his home" isn't sufficient to overcome the presumption that the mother's husband is the father in California unless the mother of the child or her husband disputes this presumption.
The document says "The reverse is not true". Why the decision about
paternity is different between a wife cheats on a husband and a
husband cheats on a wife?
Because maternity (absent a surrogacy arrangement) is almost never in doubt, while paternity is often in doubt.
Furthermore, it wasn't possible when these doctrines were formulated (centuries ago) to determine paternity reliably in all cases anyway, at least at an affordable price. Cheap and reliable paternity tests, that can be used in pretty much any circumstances{1}, have only been possible for less than forty years, which is why a case like Michael H. v. Gerald D. didn't come up until then.
{1} There have been particular cases, for example, when mother and father are both white and a child is at least partially black, where it has always been possible to do so (although even that scenario isn't 100% accurate, as illustrated by a famous historical case in which both parents had a modest amount of African ancestry that wasn't visible phenotypically). Similarly, there was the scenario of @MartinBonner where husband "was away at sea/war at the time conception would have had to occurred". Later on, blood types could disprove paternity in some cases, but not prove it with any certainty. There is a quasi-magical process described in the Old Testament for resolving such disputes involving the wife drinking a semi-poisonous liquid. In the Roman Empire, those cases were resolved by the husband who had a right to commit infanticide if he wished. In modern times, something close to the existing legal process has usually been available, complicated in certain eras by criminalized adultery, "heart balm" civil actions, and fault based divorce.
