I don't know why you're referring to second order logic in this regard, but

for all M, if M is a model (i.e. structure) for the language, then M is a model of P

can be taken as saying that P is logically valid.

Some people use 'tautology' to mean 'logically valid'. But other people reserve 'tautology' for those formulas that are logically valid in particular on account of their form regarding the connectives in the formula.