Does 'to prove' necessitate truth? [closed]

The crux of your question seems to be:

I find it unclear ... whether “demonstrating truth” requires something be true. … If I read that someone proved something, should I trust that their finding is true?

If you consider the author reliable, then yes. When the author says someone proved something, the plain meaning is that the author considers someone to have confirmed something by means of a test:

She proved God doesn’t exist.

This means the author accepts the proof. But context is important. For example, there are many proofs of God’s existence or nonexistence, and also many counterproofs. In context, one can meaningfully write:

She proved God doesn’t exist. Her proof was discredited later when a flaw was found.

In this context, the author doesn’t accept the proof.

For a good confirming definition and numerous examples of prove in context, see the Oxford dictionary entry.¹

What is proven, what is true, and what is real are different things. When logic is used to prove a statement, that means that the premises and the arguments justify the conclusion. It does not mean that the premises are true, and so does not mean that the conclusion is true. It only means the conclusion is true when the premises are true. It also does not necessarily mean that a thing being argued about is real.

All blue cats are hyperintelligent. (True, but this does not mean there are any real hyperintelligent blue cats.)

My cat is real. (True, but this does not mean my cat is true.)

This is a true map of the territory. (Means the map does not lie, but does not mean the map tells you every real fact about the territory.)