Greedy proof: Correctness versus optimality
You can use whatever proof method you want. Proofs aren't even limited to existing patterns such as "greedy stay ahead" and "swapping". Indeed, in some cases, such as the greedy algorithm for maximizing a submodular function over a uniform matroid, the proof consists of adding together a bunch of inequalities expressing the fact that the random choice was (greedily) optimal.

Usually the proof that a greedy algorithm works compares itself against an optimal solution, though when proving approximation guarantees, it could be enough to compare the greedy solution to the theoretical maximum (a case in point is the derandomized version of the random 3SAT algorithm).

Also, I suspect that "correctness" and "optimality" mean the same thing.