In your view, "spelling check" is applied to individual words, to see if they appear in a fixed dictionary (see my other comment about difficulties with choosing this "correct" dictionary in reality, though). I can imagine this view is inviting for programmers, because it's easy to implement, but I doubt anyone else finds useful a definition that says there are no misspellings in "Their coming too sea if its reel."
In my view, "spell check" applies to utterances and roughly means "all words are spelled as per the norm of the language; I can send this document to my boss/customer and they won't laugh at my spelling." It is a more user-centric view, and more complex too, because it covers intent and norms, as opposed to the comforting lookup table for a few hand-picked strings. Modern spell checkers make heavy use of statistical analysis of large text corpora, to reasonably approximate context needed to model such intent.
Once we agree on the terminology, I believe we are in agreement, so let's not split hairs. The "correctly spelled" sentence under question comes from the Wikipedia article on spell checking, by the way.