perhaps the only distiguishing thing would be age. it tends to work in your favor sometimes. if you are 25- 30 and call yourself an amateur- it is probable that you've had extensive work with various teachers and are ready and able to take on more difficult works. whereas - a student even of first or second year college might still consider themselves a student.
this is only my way of seeing things. as i see it - when you are able to take more matters into your own hands and perform things to a juries acceptance after practicing without a lot of help - you are ready to jump a hurdle into 'performance.'
usually students start out with the idea of simply 'playing the piano.' then, you take a few classes and say 'hey, this is really fun - i want to perform at recital.' when you overcome nerves and get into performance - then you say 'i want to perform more' - but as an amateur you can't get any further without more education.
i'd say education helps you cross the hurdles unless you are really gifted and talented and take yourself to new levels. perhaps professional would indicate having worked toward a master's or doctorate degree in piano performance. although, having done this myself - i feel outdone by many bachelor degreed students at west chester. i think the teacher's take into account the other things going on in your life. for me, i had a family to also care for. if you do your personal best - imo - you can call yourself whatever you want. that is why i have the username 'pianist imo.'
one thing i can't help noticing - is that concert artists just go from student to concert artist around age 9 or 10. this is simply disgusting. but, that's usually the way it goes! and, that a piano teacher can be one, two, or three of the above.
Topic: What kind of pianist are you???