Definition of a fair ball

[TheWhiteShadow]

Just trying to make sure I understand completely. Please bear with me.

2-5-1-f: A fair ball is a batted ball which while on or over fair territory, touches the person of an umpire or player, their clothing or equipment.

Does clothing/equipment need to be worn by/attached to player/umpire? For example:

a) On a bunt down the base line, the infielder's hat falls off while he is attempting to field the ball and the hat hits the ball in fair territory which then rolls into foul territory before passing/touching the base (and nothing else touched the ball other than the hat)

b) Similar as above but runner's helmet falls off (unintentional) and hits ball in fair territory before it goes foul

c) Another bunt, umpire takes off his mask when following the ball down the line but in doing so he accidentally loses his indicator and it hits the ball in fair territory and then rolls foul

Are these fair or foul?