For full-game, timing starts when your character is selected. I've now made a list of the start of all character select animations. This is what I time off of, since some can happen before the box starts to close. Bass sometimes only moves like a pixel forward on the first frame so it can be hard to see.

Mario: Jumps off the ground Luigi: Wall appears to the right Link: Faces towards the screen Samus: Faces towards the screen Simon: Wall appears to the right, gets whip Mega Man: Transforms Bass: Inches to the right very slightly Bill: Faces away from screen, legs apart Ryu: Wall appears to the right Sophia: Jason appears from the seat