IMO there are basically 2 ways for a horse to be considered an Appaloosa- or appaloosa.
One way a horse is called an Appaloosa (notice the capital "A") is by its bloodlines and where it is registered-- the original/most common place Appaloosas are registered in the Appaloosa Horse Club. There are specific rules about bloodlines, and one parent has to be "colored"-- "color" is preferred in registered ApHC Appaloosas, but not required.
There ARE registered Appaloosas that look like your gelding (who IS very cute BTW). But grey/fleabitten/speckled coats are not part of being an Appaloosa-- its just that solids and/or greys with the right pedigree can be registered Appaloosas because of their parentage.
The other way a horse can be called an appaloosa (notice the lower-case "a") is by being appaloosa-colored/patterned. many people see a leopard or spotted blanket or varnish roaning and automatically call the horse an appaloosa. This is because of their color/pattern, not their bloodlines.
Unless you can track down info on your gelding's parentage, there is no way of knowing if he could be considered an Appaloosa. If he doesn't have Appaloosa-qualifiable parentage, then he isn't one. He also would not be an appaloosa, as in, a horse called appaloosa because of his color-- because fleabitten or dappled grey is not an appaloosa color or pattern.
This does not mean anything bad about your trainer-- he might have been told the gelding was an appaloosa and was just passing on the info. A good trainer is a good trainer, pretty much regardless of his ideas about breeds or colors. Heck, the best repro vet I can use in my area insisted, for years, on calling one of my obviously characteristic, roaned-to-mostly-white registered Appaloosa mares a grey every year on her Coggins. I couldn't convince him otherwise. All that mean is that his idea of what was a grey was wrong-- meant nothing about his talents as a vet! I sure didn't quit using him or have a changed opinion of him because he called Tabby a grey-- he went to school to be a vet, not a coat color or breed expert. LOL. And he was an awesome vet.