Does a game-loss triggered by meeting the conditions presented by CR 121.4 have a particular name, or rather is there a technical (or slang) word that indicates a loss via this condition?
Yes, the slang term is "to deck" or "to deck out"
You can use the term transitively or intransitively, i.e. you can say "I decked (out)" meaning you lost due to drawing from an empty library, or "I decked (out) my opponent", meaning you won by milling your opponent to death.
Personally, I've only ever heard "to deck", but apparently both are used.
The actual event of losing the game is getting decked. You may also hear the term "mill". That term originated with the card Millstone, which moved cards from a player's library to their graveyard, and until recently it was an informal term for removing cards from a library, and could be ambiguous as to whether it referred to moving the cards to the graveyard, drawing them, exiling them, etc. However, in M21 (released in 2020), "mill" was made a keyword specifically referring to moving cards from a player's library to their graveyard. Thus, while forcing a player to draw cards does remove cards from their library, it is not milling. Milling does not directly cause a game loss; only when the player is required to draw from their empty library do they lose. However, decking and milling are still closely connected, since milling sets a player up to deck themselves.