In the proof above, we made a deduction with an inconsistent set of premises, namely the premise that the square root of two is a quotient of integers, and known facts about numbers.

I think your third remark is an odd way of phrasing what's going on, and I think it's still odd if you change "premise" to "statement." The contradiction you have is between having n1 and m1 not both even and then having n1 and m1 both even. It's a contradiction between two steps in the proof. That's the contradiction you point to, and the one that I think most people who understand the proof would point to. But at the end, you claim the contradiction is between your opening premise and all unstated mathematical facts.

There isn't a unique way to talk about the logic symbolically, but we're not butchering things much by giving it the broad structure:

1) Suppose the square root of 2 is the quotient of integers

2) Then it can be put into a simplest form.

3) Then the simplest form is not the simplest form.

4) Therefore the square root of 2 is not the quotient of integers (contradiction from 1-3)

Here, it's clear that the contradiction is between statements 2 and 3, and not between 1 and other unstated statements.