Unfortunately, I found it incomprehensible when just visiting the site.

Also, when there's nobody else in the matchmaking queue, I can't start a game, and there's no obvious video that shows me what I'm supposed to expect or how to play.

Clearly, there's lots of interesting graphic design and animati…

Thank you for the reply and giving an helpful and insightful explanation! 

Generally, you will just say “changed to state X at step number Y” and call it good. For the “jump” animation, because the movement replication and the FSM replication happens with the same step numbers, It'll still sync up. (Unless you extrapolate, in which case any user action will cause “jumps” …