What's the right term for the list of "X killed Y with Z" messages in a multiplayer shooter?

This is known as the killfeed.

This term has been widely used throughout many different multiplayer games, including old ones such as Quake, Counter-Strike, Starsiege's Tribes, etc. For the etymology of the word the origin is unknown but it has likely started off with Quake/Quake II.

Some examples of killfeed from the games you have mentioned:

From Giant Bomb:

A standard in nearly all competitive online shooters, the Kill Feed is the live on-screen update of the kills in an multiplayer game, providing updates as to whom is killing whom, as well as other important events.

I've never heard of "killfeed", but then I haven't been into games in a while. In Quake (the first one) they were called obituaries.

See here for the list of Quake's obituary messages, including such classics as "X rides Y's rocket", "X accepts Y's shaft", and "X chewed on Y's boomstick".

In some games, these messages are known as "Obituaries".