Well, no. The idea of getting paid for it as the key to understanding a "professional" is relatively new. A profession is "a calling requiring specialized knowledge and often long and intensive academic preparation." (The other fields of endeavor were crafts or trades...but somewhere along the way in the late 20th century, the term "profession" morphed.) The professions were those requiring study and theoretical knowledge (theology, law, medicine).

Amateur is one who does not do something as their vocation and is frequently untrained. The lack of training, the lack of pay, and the doing of something for fun rather than vocation are all equally important. (But see Kevin' note above about the way we define terms like "professional" and "amateurish"...)