The correct spelling is trainer as the noun for a person who guides another in learning a new skill or who is responsible for the physical workouts of an individual or team. It is also the name for those who work with animals, particularly those animals that are taught specific tricks or skills.
This word is also the name for sneakers or tennis shoes in British English.
The confusion in spelling stems from the fact that this word has taken an action and made a noun form for an occupation. Most other words that do this end in –or. For example, supervisor, commentator, and counselor. There is not a specific reason why this word does not conform. One must simply remember that trainer is spelled with an -er.
Trainor is a surname, however.
Examples
“You have to give him a lot of credit,” trainer Nick Zito said of Baffert. “You know how good you have to be as a trainer to do that.” [New York Daily News]
A Seaquarium spokesman has confirmed to New Times that trainers will no longer be allowed to perform in the tank with Lolita as a result of the investigation by the Occupational Safety and Health Administration (OSHA). [Miami New Times]
“Shoplifter yesterday tries on £90 trainers in city centre shop, puts his old ones in shoe box and leaves wearing new trainers.” [Manchester Evening News]
But when Meghan Trainor was a teenager growing up in Massachusetts, she felt shame that she wasn’t skinny. [Daily Mail]