I don't see anything wrong with using "cartoony". It's clear, effective, and to the point. If someone approached me for an image and said "I don't want anything too 'cartoony'." I'd have no problem with that and would immediately know what they meant.
It may also be a good idea to provide examples of styles, such as -- "I'd like something along the style of Archer rather than South Park or The Simpsons. Or something closer in style to Black Dynamite as opposed to Rick and Morty or Bob's Burgers."
Now, if you are trying to convey to someone that what they've already created falls into a too "cartoony" category, and you are trying to not be offensive, you might use different phrasing, such as.... "This is a bit too playful." Or "I'd prefer something more realistic without feeling like an animated character." Or "I'd like to go in a more life drawing direction rather than comic or animation."