That vs which confusion

I'm pretty clear about when to use "that" vs "which," but I often come across stuff (to be intentionally vague) that often has "which" when it should have "that." So I strike out the word and replace it, though sometimes I don't want to be a killjoy, so I leave it in, especially if the screed is several pages long and I want to vary the style. I'm not a style editor, though someone tried to make me operate in that manner, but I feel that if I keep correcting every misuse, it'll seem sort of crazy and monotone. So yes, I purposely am incorrect sometimes for the sake of keeping the peace and offering some diversity in a sea of hyper-functional sentences and concepts.

Anyway, there are a lot of resources online that explain the difference between "that" and "which." Basically, "which" is used with a clause, a subset that explains the main subject of the sentence. "Which" is a "nonrestrictive modifying clause...that adds extra or nonessential information to a sentence. The meaning of the sentence would not change if the clause were to be omitted." In fact, usually people use "which" with the sentence I just quoted from the University of Illinois; they would say "which adds extra..." instead of the correct "that." So here's an example of correct "which" usage:

The ramshackle house, which is down the block, is scheduled for demolition next week.

Essentially, the "which" section could be taken away and it wouldn't affect the integrity of the sentence. It's like an added comment to further describe the house, which is why the U of I calls it an "adjective clause."

Then there's the kind of sentence that I usually see, even by people who have lots of publishing experience with impressive titles that they display proudly on their business cards:

The house which is down the block is slated for demolition.

It should be:

The house that is down the block is slated for demolition.

In that case, "down the block" is an important piece of information, thus "that" is used, and the segment isn't set up to be separate, which is achieved with commas around a "which" clause. The U of I calls "that" a "restrictive modifying clause" because it's essential.

Actually, those definitions weren't invented by the U of I, but I like their explanation and the fact that their page isn't loaded down with ads that slow down my computer, which is common with popular grammar sites.

So, moving forward, I hope people use "that" and "which" correctly. It's not like the world is going to end, but still.