I kinda like how the second option sounds more, but that's just me. This could also have some variations, like point based instead of just asteroids, or having a sub-mode with a bigger goal but things get gradually crazier.
Just to throw ideas to the wall, I don't know how hard would it be to implement or code, but when I think of Asteroids, the most fun idea I get is a team match based on points.
Two groups of players compete to get the most points (by blowing up asteroids, enemies, maybe other players), but if a player is destroyed the team's score gets a hit. This way it could be a mix of a cat & mouse game while also having to blow up other things to improve the score, you couldn't take your attention from the other team but those big juicy asteroids are flipping around for your attention.
To avoid a team ganging up on a specific member of the other team, maybe it could leave behind a power up for its friends so it can scare them and to try and get a desperate upper hand, or if a team gets much of a head, enemies would start aiming at them more.
It could go either as a deathmach, or as a time based thing. In case of the second, there should be some sort of delay between the death of a player and it respawning.
I always liked to imagine stuff like this, but it's probably too complicated to do lol.