>And "the given word" in that particular example means "well" and is spelled G R E A T. G R A T E is a misspelling of that word.

"Grate" is a real word, and it is correctly spelled. In fact, within the purpose of the joke, it's even correctly used! But even if someone were to write that sentence out with no joke meaning, because perhaps they had learned English as a second language purely phonetically and were just trying to write things as they sounded, it'd be a grammar issue not a spelling one. Same as more common IRL hiccups like their/there, or its/it's. We even have other words like in the English language specifically to describe that in turn, like "homonym".

>Your position doesn't make any sense when you boil it down.

No, it's your position that makes no sense. You are effectively arguing that the word "grammar" shouldn't exist! There is in fact an objective difference between mechanically misspelling words and incorrectly using a homonym.

>I write some word as some sequence of letters. Whether it's correctly spelled depends not only on how that word is spelled, but how all other words, completely unrelated, are also spelled?

As I said, you're free to invent your own special snowflake definitions. But what you are writing is not in fact the shared definition at all. You for some reason are very determined to conflate "spelling" with "grammar". I linked you a few major sources, but this is not an area of contention, it has been consistently used for a very long time including in computers. It's even had plenty of attention over the decades. I still remember when a grammar checker was added for the first time to Microsoft Word and the debates about its quality (or lack thereof). There are even whole UX patterns around this, like coloring the squiggly lines below writing differently depending on if it's a spelling check error (commonly red) or grammar check (often blue). Precisely because grammar checking is harder and has often been iffier many people will disable it but leave spell checking on, because they're confident enough in their grammar and don't trust the computer, but don't want to accidentally post or send a message with "great" or "grate" as "graeyte".

Edit: in reply to wat10000 doing the 'ol virtual "good day to you SIR!" below:

I said it was a snowflake definition, given it's completely contrary to every dictionary and historical usage. I didn't call you yourself a snowflake.

And what's actually really fucking infuriating is when people like you simply refuse to use standard, shared dictionary definitions of words and widely used established software tools in your conversation and then further refuse to acknowledge it when corrected. And also refuse to engage with any substance and instead storm off in a virtual huff. You could have just gone "right I meant grammar correction, present grammar correction really kind of sucks and that's what I think people need most vs spelling correction" and that'd be that.