The term "coach" refers to "the direction, instruction and training of an athlete." Most tennis players have many coaches who have participated in various capacities throughout their career, but all coaches are not equal.
A coach may provide lessons and advice many hours a week, for many years, to players who also participate on drill courts, high school tennis, Zonals, Intersectionals, etc., with other coaches. While all of these coaches, by strict semantic definition, have a right to say they "coach" this player, when used as a marketing tool. It can be a misleading and disingenuous practice that exploits players and parents who judge the success of a coach, (and make buying decisions) based on the players they have developed.
If a tennis coach refers to a player as someone they "coach," it's reasonable to assume that they have been the most significant primary coach of this player. It might also mean, however, that they have walked on the court together no more than once or twice. I think I have heard over 20 pros claim to have been John McEnroe's coach over the years.
While most tennis teaching professionals abide by this unwritten fairness code when it comes to claiming credit, several are notorious for disregarding it completely, therefore I suggest "Caveat Emptor" when choosing a coach based on their player resume.
Steve Kaplan is the owner and managing director of Bethpage Park Tennis Center, as well as director emeritus of Lacoste Academy for New York City Parks Foundation and executive director and founder of Serve & Return Inc. Steve has coached more than 1,100 nationally-ranked junior players, 16 state high school champions, two NCAA Division 1 Singles Champions, and numerous highly-ranked touring professionals. Many of the students Steve has closely mentored have gone to achieve great success as prominent members of the New York financial community, and in other prestigious professions. In 2017, Steve was awarded the Hy Zausner Lifetime Achievement Award by the USTA. He may be reached by e-mail at StevenJKaplan@aol.com.