There are, as you say, many different logics. And sometimes they differ as to whether a particular form of inference is valid. The form you have chosen there is called modus tollens and is valid in nearly all commonly used logics. There have been some claimed counterexamples to this form, but none that have found general acceptance. As far as I can see, this form would fail only in a paraconsistent logic that tolerates true contradictions.
A better example might be something like this:
- Not everything has the property F.
- Therefore, there is something that lacks the property F.
This argument form is valid in classical logic, but not in intuitionistic logic. The difference here may be attributed to a difference in what the intuitionist understands by a claim that something exists. Classical logic permits non-constructive existence claims: something of a particular description exists, but I can't show you what it is. Intuitionistic logic is more fussy and does not permit a claim of existence unless it is accompanied by a constructive warrant.
Another example is this:
- P is true.
- Therefore, P or Q is true.
This form is valid in classical logic, but is not accepted in some relevance logics, because the Q in the conclusion is irrelevant to the premise.
It is common to understand classical logic as accepting an argument as valid if it has no counterexample, i.e. there is no possible case where the premises are all true and the conclusion false. This may also be expressed as saying that in a classically valid argument, there is a guarantee of truth preservation from premises to conclusion.
For many other logics, it is possible to understand them as preserving some other property. Intuitionistic logic preserves constructive provability or warranted assertability. Probability logic preserves high probability (approximately). Relevance logic may be thought of as preserving the integrity of information in a channel. Linear logic may be thought of as preserving the integrity of resource-bound interaction.
More generally, it makes sense when examining a logic to ask about its semantics. What do the symbols in this logic mean, and how are they intended to be used? The answer to this question usually provides the answer to whether particular rules of inference are valid or not. The rules are validated against their use.