Softmax is defined over an arbitrary vector of raw real numbers. Stating that those inputs are "logits" is applying post-hoc semantics to what the model is learning. One of the key properties of a softmax is scale invariance, (e.g. softmax([-1, 1, 3, 5]) == softmax([9, 11, 13, 15])) and so it is easiest to just think of it as operating on a vector of unnormalized raw scores, which is the more colloquial definition of logit.

(also, log(p) is not the formal definition of a logit)