When does a threat become a crime?
When it is an unreasonable request (i.e. a demand) that is unjustified in the circumstances (i.e. unwarranted) in order for someone to gain or lose something.
I will only offer a general answer according to UK law as there is insufficient information in the Richard Barnard link (presumably as it sub-judice and not all the facts are in the public domain) and the Compound Labs case seems to be American.
The united-kingdom laws on blackmail may be found variously at:
(1) A person is guilty of blackmail if, with a view to gain for himself or another or with intent to cause loss to another, he makes any unwarranted demand with menaces; and for this purpose a demand with menaces is unwarranted unless the person making it does so in the belief—
(2) The nature of the act or omission demanded is immaterial, and it is also immaterial whether the menaces relate to action to be taken by the person making the demand.
(3) A person guilty of blackmail shall on conviction on indictment be liable to imprisonment for a term not exceeding fourteen years.
...using threat of harm to demand money, property or some advantage from another person. It does not matter whether the demand itself is legitimate (such as for money owed) as the crime can still be committed when illegitimate threats of harm are used.
Ultimately it will be for the jury to decide each particular case after considering all the facts and circumstances whether a defendant's actions are reasonable and justified or not, but the following case law gives clarity on some definitions:
I think the word 'menace' is to be liberally construed and not as limited to threats of violence but as including threats of any action detrimental to or unpleasant to the person addressed. It may also include a warning that in certain events such action is intended. [Lord Wright]
... no act which was not believed to be lawful could be believed to be proper within the meaning of the subsection. [Bingham J]
A very simplistic (and facile) way of putting it is that if you don't pay your bill the Gas Board can cut off your supply but they can't cut off your head.