If i am honest i have no idea what all of them mean. I dont know the difference between a "medium" steak, a "medium rare" steak and a "medium well" steak. I don't think I would be able to tell if I ordered a "medium rare" and they brought me "medium well" ...
I just always order "medium rare" because people have told me that this is how a steak is supposed to be prepared and that ordering "well done" or something else would mean the steak is now basically garbage.
All these doneness levels really refer to the temperature the steak reached.
Rare: 120-130°F. Bright red center. Fat isn't really rendered yet, meat generally a bit fibrous and slippery. Very juicy.
Medium rare: 130-140°F. Bright pink center. Fat is mostly soft and rendered, meat is a bit firmer now and not as slippery. Still very juicy (possibly more so than rare, depending)
Medium: 140-150°F. Pink/grey center. Fat is very well rendered, some would argue starting to render out a bit too much. Meat is firm, but still somewhat tender. Juicy, but not very.
Medium well: 150-160°F. Almost entirely grey center. Everything is getting quite firm, bordering on hard. Not much "juice" at this point.
Well done: 160°F+. Grey through and through. Very firm, depending on the steak possibly bordering on hard to chew, though that's honestly not very common.
What most people are looking for are their preferred combinations of meat/fat texture, rendering level of fat (how soft/cooked the fat is), and juiciness.
There are other factors that influence all of these aspects other than the cooking temperature, such as salt penetration (salting overnight gives you a firmer, juicier steak), but doneness is a major factor.
The people telling you well done is garbage are spouting a common opinion, but you should feel no shame ordering what you like to eat.