1

"We couldn’t put down the new floor till the plumber had finished."

"We couldn’t put down the new floor till the plumber finished."

"I wait until he finishes."

"I wait until he has finished"

I don't understand how I can choose the correct tense.

2

There is no "correct". All of these are grammatical; but with a continuing process, the perfect is more common.

So, with a point event, it is common to use a simple tense

I'll wait until he gets here.

and a perfect for a continuing event that reaches an end

I'll wait until he has finished.

But the other way round is also possible:

I'll wait until he has got here. I'll wait until he finishes.

[One other point, that you didn't ask about: with most verbs, the so-called "present tense" is not a present tense at all, but a timeless or habitual tense. So in normal speech "I wait" is used only in a habitual sense. Here you want either "I'm waiting" or "I'll wait".]

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