When does a chess clock start? Lichess.org starts the clock of each player after their first move.
Chess.com  starts the clock when the players agree to a match, before
their first move.


I think Lichess is correct, that's what we do where I play. What's the rule?
According to the FIDE rules,


  6.6 At the time determined for the start of the game White’s clock is started.


In over the board chess, it's usually Black who does this (you can't have an arbiter start all clocks simultaneously) after the players shake hands and wish each other good luck. If White isn't present, Black may start the clock anyway at the designated time; if Black isn't present, White should start his/her own clock, make a move and press their side of the clock to start Black's. (Of course, in some events, failing to show up at the designated start time constitutes an immediate loss anyway.)

So actually chess.com's method is closer to the official rules, though those games are of course not regulated by the FIDE rules.