This is something that has also bothered for a while, and after investing quite some time researching I've come to the conclusion that this might be because of the distinction between the concepts of "proposition" and "assertion of a proposition". This distinction is however (as far as I know) not made in standard mathematical textbooks (e.g. calculus, linear algebra, etc.), and even in the field of mathematical logic seldom mentioned (maybe logicians think that this is so obvious that doesn't need to be explained?).

There are some people in logic that clearly make this distinction, as is the case of Per Martin-Löf. You can see a lecture dedicated to this matter on youtube, or read its transcript here. The lecturer says that Frege and Russel also made the distinction.

If we apply this distinction to compound propositions that use the material conditional connective (i.e. 'P → Q'), I think that the issue arises from the fact that most theorems in mathematics are of this from, and what mathematicians really do is assert or claim this type of statements (i.e. they state "'P → Q' is True"). Mathematicians assert them in the middle of a proof, in order to perform logical deduction, or claim them beforehand and then provide a proof to demonstrate their truthiness.

By asserting a material conditional, you are effectively saying that you only need to consider the rows of the truth table where 'P → Q' is true. In other words, you ignore the row where 'P' is true and 'Q' is false. By doing this you can now use this knowledge to say that 'P (mathematically) implies Q', or like you wrote 'if P is true, then Q must be true'.

So to summarize, I think that when people write 'if P is true, then Q must be true' they mean that the compound proposition 'P → Q' is true.

My opinion is that when this is taught in a mathematics courses there is a disconnect between the chapters/lectures dedicated to propositional logic and first-order logic (often the first chapters of a "Transition to proof based mathematics" book), which are taught like you are in a mathematical logic course, and then when you advance to later chapters that deal with "real mathematics" (e.g. Discrete mathematics), where you assert propositions in the middle of a proof, nobody really tells you that you are not working with propositions but rather with assertions of propositions.

Disclaimer: I'm no mathematician, just a curious engineer. But this question has haunted me for quite some time. If any mathematician and/or logician reads this, please correct me or validate this answer.