As a composer myself, I tell non-musicians that I'm a Professional Composer, it's just the easiest way to describe it. Here's the thing, very, very few composers throughout history were able to make a living solely off writing music. Even now, there are a only a handful of composers in the US who are able to do this. JS Bach worked for 4 different churches, Haydn work for the Esterhazy court, and many, many, many composers throughout history made their primary source of income by teaching and not composing.
Even most of the best-known composers today teach full time and pursue composing outisde of their regular job, and yet, I'm sure no-one would argue that Frank Ticheli, Jennifer Higdon, or John Corigliano are professional composers.
That said, here are some very common ways to refer to composers at various points of their career:
Student Composer - Any composer still in school (BM, MM/MA, DMA/PHD)
Emerging Composer - Successful graduate students and young professionals outside of school up until around the age of 35.
Established Composer - People that experience a consistent amount of work / recognition; people with advanced careers.
Beyond these three points, it can be specific to the vitality of their career: world-renowned, seasoned, etc etc.
In my experience professional and amateur refers more to the mindset of the composer. Someone who is very entrepreneurial, always delivers on time, and actively pursues opportunities is someone who is professional. Someone who is lackadaisical, misses deadlines, and doesn't actively pursue opportunities is amateur. Amateur composers don't typically make it past the "Emerging" phase as they just don't put in enough work.
I hope this answer sorts it out for you.