definition of formula validity I read in some sources that valid formulas are tautologies (valid under every evaluation). In the others, I read that these are formulas that have conclusions true when premises are true. Are these just equivalent definitions because ⊨ P → Q is equivalent to P ⊨ Q?
Yes. These definitions are equivalent,  noting that not all valid formula have the form $P\to Q$, though any formula $Q$ is equivalent to $\top\to Q$.