Structure of a Pumping Lemma proof: contradiction or counterexample?
this is not context-free/regular because I can come up with a counterexample which fails the pumping lemma


You are missing that in order to construct this counterexample you have to assume that $L$ is regular. Then you apply the Pumping lemma, which yields the Pumping length $p$. Only then do you construct your (counter)example string (using $p$!) and show that it contradicts the Pumping lemma. Hence, the assumption must be false.

So while you can say that you derive the contradiction using a counterexample, the outer-most proof is by contradiction.
With the assumption, you wouldn't have $p$!

Note that proofs can be nested, so you don't need to see only one proof technique. For instance, induction proofs often use a proof by contradiction inside the inductive step.