You've misunderstood the difference of "sameness" those two works are trying to address. Knuth is not using an equivalent idea of "sameness" as discussed above. Two people each going by Philosophy of Computer Science and the Art of Computer Programming exercise notion would not always agree if two programs are "the same".

It's also a bridge too far to make a blanket statement of Knuth "already answered this" (it's also not accurate to attribute that idea to Knuth; e.g. Church did work exactly on this decades before). Meaningful discourse and mathematical analysis requires nuance and degrees of equivalence and mixing them up makes it needlessly confusing and difficult.

Sameness as "same input same output" is NOT the same as the "isomorphic equivalence" which is more difficult, strict, and abstract. In this sense they are the same if they follow the same "steps" (also isomorphic between programs), i.e. they are different representations of the same algorithm.