For sure things could've been worded/presented better, but the rules do say exactly what I presented.Someone should have told this to the designer who wrote the rules for hiding, because the line, "When you try to hide, you use the Hide action," completely misrepresents the core game loop. Your description of how hiding works is clearer and more insightful and than anything present in the core rules.
The DMG under Resolving Outcomes says "Players shouldn't just roll ability checks without context: they should tell you what their characters are trying to achieve, and make ability checks only if you ask them to" and then a few sentences later "Call for a D20 Test only if there's a chance of both success and failure and if there are meaningful consequences for failure".