It's counts for both, since the rules declare that the timer starts from when you interract with the team's door, it counts everything from that point on, the real difference with RTA and LRT is if after you click the door it continues to load instead of immediatly giving you the opportunity to choose a character. This commonly happens when you frame-lag, like in some of my other speedruns.

Oh sorry, I just play 2 times either because the first onemay be too laggy so it's not verifiable, and also because usually the online version is 2 matches to win the game. Won't happen next time!