I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula?
Some people use "tautology" in logic in a wide sense, to mean any logically true wff. But others use "tautology" more narrowly to mean true in virtue of truth-functional structure (so, "valid by the truth-table test"). So, for example, $\forall xFx \to Fa$ would count as a tautology in the first, wide, sense but not in the second, narrow, sense. (How to use 'tautology', then, is a matter of terminological preference: I much prefer the second narrower usage as it gives us a label for a special class of logical truths which we need a label for.)
They also say that $A, B \vDash \Phi$ is valid if it preserves the truth
No they don't. Or at least, they shouldn't. It is the inference that preserve truth. Thus if $A, B \therefore \Phi$ necessarily preserves truth, then it is valid. But $A, B \vDash \Phi$ is not an inference: it is a meta-linguistic statement about the inference, and says the inference is valid.
They also say that in addition to being valid, i.e. preserving the truth, your argument is sound when premises are also valid.
Fine: an argument is sound if it has true premisses and a valid inference move.
But there is another definition of soundness, which tells that theory is sound if formula provability implies that formula is semantically valid.
As you almost say, this is a definition of what it is for a formal deductive system to be sound.
Arguments aren't formal deductive systems, and soundness for arguments is not soundness for formal deductive systems. The notions can't be identical. But there is some similarity between the two notions -- both have something to do with being trustworthy (an argument won't lead you astray about the truth if it starts with truths and proceeds by truth-preserving moves, i.e. is sound; likewise a formal system encoding inferences won't lead you astray about what's a valid inference if it is sound)