I believe your “checkup” and their “routine cleaning” are the same thing.
Lots of people (myself included) refer to it as a “cleaning” because, well, regardless of anything else, that’s what they actually do to you. I don’t know anyone who goes to a dentist just to have them look but they don’t touch. They clean you, too. That’s almost always the only physical takeaway effect of one of these visits.
Also, a dentist cleans your teeth in a way you almost certainly can’t. Their practiced hands know exactly what needs to be scraped away, and they can make informed decisions on what tool to do it with and how aggressively to not cause enamel damage. Not to mention they can, y’know, actually see what they’re doing in there. So a “simple cleaning” isn’t quite as pedestrian as it sounds. It’s not something you can fully replicate by scraping around blindly with a metal pick in your mouth.