You're right that those are all espresso-based drinks, but those aren't "coffees," they're drinks that have coffee (espresso) in them. To use an analogy, calling a latte a kind of coffee is like calling a martini a kind of spirit. Generally those are drinks one would get at a cafe (or Starbucks), made by a barista, using the very expensive commercial espresso machines mentioned elsewhere in this thread. Of course, if you have the means to make espresso, you can also make those drinks at home.

The word espresso just refers to a particular method of brewing coffee (albeit one that produces a very different style of coffee from other methods). There's no special name for French press coffee because that's all it is: coffee made in a French press. Unlike a latte, it's not a fundamentally different drink from, say, auto-drip coffee, or even espresso, which, while being very different from coffee prepared in other ways, is still coffee at the end of the day.

So basically, there's espresso (from which lattes etc. are made) and then there's everything else (e.g. auto-drip, French press, pour-over, Aeropress).

I'm kinda tired, I hope this post made sense.