I've done a fair bit of beta-reading, including for non-English speakers. Oddly enough, I don't split into 'types' of criticism in my head, I think of it more as a series of levels and work at a different level depending on what type if crit the author wants.

Things That Are Wrong- either grammatical; canonical (getting ages wrong when they've been clearly stated, sating two people have never met when we've seen them on screen) or factual (you can't drive from London to Edinburgh in two hours. Really,you can't).

Things That Are A Bit Off- stylistic tics (repeating the same word even though it's used correctly every time); exposition and flow problems and flaky internal characterisation.

Things Maybe Only I Would Notice- pet hates of my own (why do so many writers use italics? learn to write a sentence properly and you don't need to fall back on formatting); inadvertent jokes (words with different meaning in Britain) and characterisation that differs from my personal view.

I put characterisation in the final level, because I know everyone has a different perspective and finding a new way of looking at characters is part of what fanfic is all about, no matter how much I'm tempted to put men crying in the first level and convince the world that it's flat out wrong.

I agree with your advice section wholeheartedly though, especially on how to remember that it's still the author's story and not yours, with the slight caveat that even asked for a spelling and grammar beta I would still point out basic factual errors as well.