What term is used for same-screen multiplayer?
If players take turns, that's called turn-based multiplayer.
If players play at the same time, that's called simultaneous multiplayer.
If players play on one screen at the same time, there's no term for the "shared screen"ness. That's just the default way for console games.
If players play on different screens at the same, there's no term for the "separate screen"ness. That's just the default way for online games.
If players play on one screen, with a portion of it under their control, that's called split-screen.
If players compete for objectives, there's no term for that - it's the default.
If players cooperate, that's called co-op multiplayer.
Examples of "shared screen" games are New Super Mario Bros for Wii and Lego Star Wars (and technically - Mario Galaxy).
Examples of split screen games (which also support online play on different screens) are Borderlands (xbox360) and Minecraft (xbox360).
The more recent games from the Lego Series use a shared screen, that splits when the players get far enough apart.
That's called "shared screen", as the players share the screen.
You don't see this term all that often, as not only are these games not particularly common, but it's also usually not necessary to make the distinction. People will say "split-screen" if the players split the screen, and not mention it otherwise.
Here are a few examples of this term in use:
GamePeople's "Shared Screen Videogames" section (GamePeople is a video game review site).
The term being used in a Unity forum thread, and everyone understanding what it means.
Steam Curators page for "local multiplayer games" listing types as "Shared screen, split screen, hotseat, coop, competitive, and every other type".
Without beating around the bush, offline multiplayer has been traditionally known as Local Multiplayer.
More recently this has been generalised (or specialised depending on the way you look at it ) to more informal terms such as Couch Play. Furthermore Split-Screen Multiplayer is only one particular way of implementing Local Multiplayer and therefore by definition is a sub class.
In some cases Local Multiplayer also refers to multi-user play on a LAN, but the term it self covers both offline and online multiplay.
Hope that helps.