Angled flat seats are flat. The correct word for what you're looking for is fully-horizontal flat seats. You still "lie flat" on an angled flat seat, ie, if the seat flat but at angle it's still flat.

Flat refers to lack of curvature or lack of bend.

(Not that airline sites or third-party sites use the correct terminology either!)

I've been on fully-horizontal seats that were not really flat (had a painful "bump" where the two parts joined!), on Air Canada CPH-YYZ, and that was way less comfortable than AA's angled flat seats. (Luckily it was a daytime flight, and I didn't need to actually sleep, I just tested out the sleeping position of the seat for future reference.) I'm sure AC's hype calls them "lie flat". (You have to read reviews to learn the truth. I've never seen another airline so criticized for their supposedly "lie flat" seats as when I searched for reviews of Air Canada.)

So be careful assuming that fully-horizontal seats are necessarily better than angled flat. Besides the sleeping angle, there are big quality differences between sleeper seats, and some "flat" seats are not really flat but have bumps !