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 $\top\to Q$.
• If you want the two statements to be equivalent, then I need to be able to talk about statements that are not of the form $P\to Q$. Fortunately, any formula (not just any valid formula) $Q$ is equivalent to $\top\to Q$. – Dave Clarke Sep 25 '12 at 19:55
• $\top$ means true. It is possible to have multiple definitions of a concept, as long as they are equivalent. Different definitions may be better suited for different purposes. I never said that any formula $Q$ can be denoted $\models Q$. I merely gave you a tautology, namely $Q\leftrightarrow (\top \to Q)$. – Dave Clarke Sep 25 '12 at 22:07