Also, like, the language “skill check” doesn’t appear anywhere in the 5e rules. The idea that an ability check that benefits from a character’s proficiency bonus due to their proficiency in a skill is a skill check and that the difference is merely semantic is fundamentally rooted in the language of previous editions. And I don’t think framing a system in terms of its previous editions is a good way to go about forming an understanding of the system. Are ability checks in 5e functionally similar to skill checks in 3e and 4e? Sure, but they’re not identical, and thinking of them that way is going to lead to a warped understanding of the 5e rules system.
I simply find "Make a dexterity ability check with your proficiency modifier and any other possible modifications to stealth" to be just a tad wordy. Because it's not just the ability modifier, it's the proficiency, possible expertise, magic items, spells and magical effects that could also come into play. On the other hand if I say "make a stealth check" people know what I'm asking for.
It's much, much easier to tell a newbie to roll a D20 and add their skill bonus. I don't really care if the book ever calls it out, I'm running the game in a fashion that I think makes the most sense and is easiest to understand for my players.
But this horse was dead a long time ago. Have a good one!
