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.
Yeah, mom liked her steak well charred. Once we were old enough to man the barbeque ourselves the quality in steaks improved, though she still insisted we burn hers black.
Sirloin or rump, I dont really mind gotta be cooked just right, room temp slab of meat, ten mins before it hits the pan, bathe it in some good quality olive oil. Give it a nice little massage. Then get the pan real hot, full burn, usual salt and black pepper, coarse grind setting and slap that bad boy on there for about 2 mins. Dont touch it only apply some light to medium pressure to get a nice sear, stay on full burn, dance to that sizz. Flip over can do 1 min for rare, 2 mins for med rare, then I grab it and sear the sides off a bit then stick it on a plate to rest for 5 mins. Usually have fried onions mushrooms and if im feelin fancy, some fried asparagus, tastes real good cooked in the steak juices and on the verge of gettin charred. Usual chips or whatever to go with, but the steak is the star of the show.
Edit: <COUGH> sorry, my response got stuck in my throat. It was a little under done.
As I was saying, it's difficult, because when I'm out I have to remember how the restaurant does it. I've been in places where medium ends up burnt to a crisp, and others where if you order it medium, they walk the steak out to you still chewing on some grass (I exaggerate, but you get the idea).
Shouldn't have to worry about it, but not every place gets it the same. As long as they are consistent, I can deal with it.