The decision precedes the stroke
Rule 13 permits leaving the flagstick in the hole. A ball hitting that normally positioned flagstick carries no penalty, and the ball is played as it lies. Deliberately altering it away from centre to obtain an advantage has different treatment.
Imagine a long putt reaching the hole, striking the flagstick and finishing a few centimetres away. The rebound may disappoint the player, but the contact alone does not require a penalty or a replay. The shot’s outcome remains part of the hole.
Do not infer the choice from the final frame
Another player may remove the flagstick for their own stroke immediately afterwards. A still image of the now-empty hole cannot show what the first player decided before putting. Follow the flagstick through the original stroke and rebound.
This also explains why two golfers in the same group can make different choices. Their preferences do not alter the ordinary rule for the other person’s completed shot. Deliberate attempts to affect a moving ball raise separate questions, so avoid extending the simple left-in-place example to a late intervention. The useful observation is the flagstick’s status when the player made the stroke.
Questions answered
Does a putt hitting the flagstick automatically need replaying?
No.
Must everyone in the group make the same flagstick choice?
No; the choice concerns each player’s stroke.
Sources & further reading
Official references for the subjects discussed. Service terms and device software can change.