You can be hourly and exempt. You can be paid a fixed salary and be non-exempt. There are also many salaried people who fill out timecards.

Edit to your edit:

> my "often accurate" generalization here is not accurate often enough to be worth saying, which I'm willing to believe. No level of detail in a HN comment can substitute for hours of research and talking with a lawyer

Your generalization is frequently incorrect, because billing by the hour is extremely common. The DOL has a very readable website that explains exemptions. https://www.dol.gov/agencies/whd/fact-sheets/17a-overtime