It depends on the law
For example, the NSW Crimes Act 1900 s18 defines murder and manslaughter:
(a) Murder shall be taken to have been committed where the act of the accused, or thing by him or her omitted to be done, causing the death charged, was done or omitted with reckless indifference to human life, or with intent to kill or inflict grievous bodily harm upon some person, or done in an attempt to commit, or during or immediately after the commission, by the accused, or some accomplice with him or her, of a crime punishable by imprisonment for life or for 25 years.
(b) Every other punishable homicide shall be taken to be manslaughter.
(a) No act or omission which was not malicious, or for which the accused had lawful cause or excuse, shall be within this section.
(b) No punishment or forfeiture shall be incurred by any person who kills another by misfortune only.
It is possible that the acts you describe could be prosecuted as either murder or manslaughter. For murder the prosecutor would need to prove that they are an "act by the accused ... causing the death charged, was done ... with reckless indifference to human life" - the hard part is the causal link. For manslaughter it is clear that the act "was ... malicious" and that there was no "lawful excuse" - some cause and effect would need to be demonstrated still.
However, the prosecutor has a more certain option under s31C:
31C Aiding etc suicide
(a) a person incites or counsels another person to commit suicide, and
(b) that other person commits, or attempts to commit, suicide as a consequence of that incitement or counsel,
the firstmentioned person shall be liable to imprisonment for 5 years.
It is important to look at the legal definition of a crime to determine if a given set of circumstances meets all the required elements.