Yup, a lot of people tend to refer to any non-zero-sum game as a "prisoners' dilemma", which is frequently wrong, so it's good to point out that that's not correct. This taxonomy is perhaps doing a little too much; I think that most of the time you really just need Prisoners' Dilemma, Stag Hunt, and Chicken (anti-coordination) as your three main examples for comparing other things to...
Is one of those games equivalent to snow drift?
Chicken and Snowdrift are two names for the same thing, yes.
A major limitation of "prisoner's dilemma" is that it's a single-play game. Most real world games are iterated with information passing cycle to cycle which radically changes the optimal strategy.
You had my interest but this is too hard to read when you dont define any of the variables, or explain how a payoff matrix works before jumping into equations