No problem! Obviously the OP can chime in if their game idea is different.
It’s complicated because MUD sounds like such a general term, but over time it became synonymous with something more niche: text-based games that in some way resemble the earliest multiplayer online game: MUD; also known as Essex MUD.
The main key to me is “text-based,” most of them didn’t use any type of character art like you’d see in Dwarf Fortress or nethack. The majority of what you do in an MUD is read sentences someone wrote of your current location or what you’re doing. It’s a lot like dungeons and dragons or interactive fiction.
You’re totally right though, in the most general sense, a lot of online games could fit into the general “Multi-user D-” term; in the early days of graphical MMOs, people called games like Ultima and EverQuest “graphical MUDs” for lack of a better way to describe them. Now we have “massively multiplayer online” to help.
So in the broadest sense, Fallout Online is a Multi-User something, but probably doesn’t fit with most people’s idea of a MUD.