Ok, but this is a technicality. Spell-checkers have slowly evolved into grammar checkers and what people really want is error correction. Whether people call it a spell checker a minor language issue (and the kind of things humans do all the time).

When teaching for your dictionary, ask: "is it obvious what they mean if I'm not being pedantic?"

We expect different outputs in these two cases, though. A wrong word choice is usually accompanied by a hint that another word may have been intended, while a wrong spelling can be unambiguously marked as a mistake. These two behaviours can be turned on and off independently, and they need two different labels.

Agreed. "Dessert" vs "desert" - mistaking these two is often not a grammatical error (they're both nouns), but is a spelling error (they have quite different meanings, and the person who wrote the word simply spelled it wrongly).