This is because in older slides, it was first assumed that the considered Kripke structures were defined on the same set of variables. If that is so, then both definitions are the same. Later is was then generalized to different variables. It turned out that many students missed the later generalization, so that the slides were updated in that the general case is considered from the beginning.

So, why should we compare Kripke structures with different sets of variables as all?

Well, recall that it is possible that several states in a Kripke structure may have the same label. That should be surprising for a deterministic system since those states should behave the same. However, it is often the case that there are local variables in system descriptions that are not visible to the outside, and that leads to these states with same labels, but different transitions.

To encode them by propositional logic, we have to introduce further variables so that with the further variables every state has a unique label. But then, we should not take these new variables into account for comparing states, since for another "similar" structure, one might have chosen other variables. So, we should ignore the new variables and should only consider the "observable" variables. That finally leads to the definition that is now seen on the slides.

Does it help?