On the difference between $\vdash A\to B$ and "If $\vdash A$, then $\vdash B$"

I thought that I can interpret $\vdash A$ ($A$ is derivable without open assumptions) as "$A$ is true" without causing any harm.

Sorry, but no. $\vdash A$ is a much stronger statement than simply "$A$ is true". As you point out, it means that $A$ can be derived without any assumptions, and that shows that $A$ isn't just True, but that $A$ is always true: $A$ is a tautology!

Consider the statement "It rains". This statement could be true or false, depending on where and when (basically, in what world) you evaluate it.

This is quite different from a statement like "it rains or it doesn't rain": that statement is true no matter what, and we call it a tautology.

In propositional logic, the first statement would be like $P$, and we cannot prove $P$ from no assumptions whatsovever. But the second statement is of the form $P \lor \neg P$, and that statement we can prove without any further assumptions. That is, where $P$ is an atomic statement, we have $\not \vdash P$, but we do have $\vdash P \lor \neg P$

If for your statement $A$ it is the case that $\vdash A$, then $A$ is like the second statement, not the first.