That definition (as interpreted by you) would appear to, at the very least, preemptively disqualify any proof by exhaustion ("it's true because we brute forced all the possibilities and didn't find any counterexamples"), which is a perfectly valid proof method.

https://en.m.wikipedia.org/wiki/Proof_by_exhaustion

"All the steps are valid but it doesn't explain anything and I'm dissatisfied" is not an argument against a proof's validity. Sometimes the argument is just "A implies B implies C implies D implies E implies P, QED" and as long as you are convinced that each step is valid, you've got to take the proof itself as valid, even if the gestalt isn't satisfying.

That is not Stanford's official definition. That is the definition from one of their Computer Science lecturers.

I can't speak to it's accuracy but I would imagine better sources exist.

That's a misc slideshow from a class, not a formal definition.

"A mathematical proof is an argument that demonstrates why a mathematical statement is true, following the rules of mathematics."

yes, and?