Someone can become certified as a pilates instructor, so that's probably your best bet. There are some nuanced differences between the three words, though all three make sense, instructor seems to be the most commonly used as well.
Instructor is the broadest of the three terms. You can be a pilates instructor, a math (or maths) instructor, a cooking instructor, or even a Super Mario Brothers instructor. You often see pilates instructor or yoga instructor (or teacher). Even a piano instructor (or teacher).
Trainer is most commonly associated with physical training, and would apply to pilates as well, though it's just not as common. A trainer, in most sports, is someone who works on core aspects of something. In American Football, a team might have a strength trainer, and a kicking trainer - someone who teaches specific tasks or processes.
Lastly, coach can mean either of the above, but is usually someone who has experience in the activity, but doesn't actually teach by example. In pilates, or yoga, the instructor is usually doing the same thing the class is - teaching by example. A coach is more likely to just tell you what you should be doing. A football coach doesn't take the field with the team. A boxing coach (usually) doesn't get in the ring for punches. There are exceptions, but they are just that... exceptions. A chess coach will probably play against you, though they may no longer compete professionally.
In short, instructor is the way to go.