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.
gaming.stackexchange.com/users/18916/korrasaur
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".
gaming.stackexchange.com/users/106852/narb
In some games, these messages are known as "Obituaries".
gaming.stackexchange.com/users/27235/Eric Finn
The area where kill notifications are posted is called the "Killboard" in many games including TF2 and Eve Online.
gaming.stackexchange.com/users/26042/Lawton
Killfeed is the correct answer
however it is a term much that is much older than Cod and probably dates back to quake
, I know I knew of it long before COD exsisted.
The results you are seeing are biased due to popularity of search terms and lack of inter-generational gaming history.
The first COD game was in 2006 and the term kill feed was widely used before then.
Google dates the term 'kill feed' back as far as feb 2004 but has no prior data to that.
The term 'killfeed' dates back to 2010 where we also see a spike in usage of 'kill feed'.
This corresponds with the release of COD: Black Ops.
Another example of this bias from an article saying Starsiege:Tribes has a notable kill feed with custom messages e.g. 'Player A blasts Player B with the Spinfusor'.
However the quake series did this first including custom environmental deaths 'PlayerName Melted' when you fall in lava for example.
Its possible its as old as Doom!
gaming.stackexchange.com/users/106814/KoolMintCookies
I usually call that a "killfeed" as well, but have also heard it referred to as the "log".
This makes sense, as it documents what's been happening in the game.
gaming.stackexchange.com/users/108003/DCShannon
It's probably obvious from the title, but I'm specifically thinking of messages like   ProGamer64 [image of shotgun] L33tCamper   that appear in the bottom right corner when a kill happens, then fade away after a few seconds.  Is there a standard term for these messages or the list of them?
What's the right term for the list of "X killed Y with Z" messages in a multiplayer shooter?
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<terminology>
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gaming.stackexchange.com/users/99431/Ixrec
This known as killfeed. In Quaker they were called obituaries. The most common term that is used to describe the action in question is killfeed.
The most used term is killfeed. While few users suggest it is called obituaries. However a user is calling it killboard.