A Coach's badge will help you identify their Specialties. The creature crest provides the Racer creature(s) type they specialize in working with. There are 2 types of Coach Specialty.

1. Creature Specialty

Rookie and Veteran coaches buff a particular Racer Creature 'family'. For example, if the badge displays 'Dolphin', the particular coach's buffs will apply to 'Creature trait: Dolphin', 'Dolphin (Pink)' and 'Dolphin (Wizard)'.
Creature Specialty Visual

2. Division Specialty

While the badge of an Elite coach will display their Creature Specialty, these coaches have the added bonus of a Division Specialty (which trumps Creature Specialty).
This means their buffs will also apply to the entire division that their displayed creature belongs to.
Division Specialty Visual

Legendary Coaches

Legendary Coaches have multiple Division Specialties. Firstly, Legendary coaches buff all racers of the 'Mythic' Division. On top of that, the initial owner of a Legendary Coach will get to choose a Division Specialty upon minting, for example 'Arctic'. This chosen Division Specialty is change-able anytime for a 50K $PEARL fee.

Legendary Racers

Legendary racers always receive the same buffs as their coach's Creature or Division Specialty