I think that, in general, math academics who have spent 10+ years working in some field will be hard-pressed to accept a person with a fresh B.S. as a mathematician, simply because they feel like there needs to be some "requirements". You yourself have fallen into that trap simply by saying that since you have a published paper you are a mathematician and no one without a published paper should be considered as such.

In general, a human being is going to judge themselves to be worthy of some title and then judge other people for that title based on their own experiences and knowledge. I think that we need to be careful of assigning requirements to titles for those titles that aren't as important. For example, there are clear requirements to be called "doctor" or even within that field "surgeon". This is because it's life-or-death and you need to know a certain amount to be called that. There are clear requirements as well for things like "police officer", "lawyer", etc. although for something like "lawyer" there is a literal "bar" set to determine the title. For mathematician, I think that many researchers consider themselves a mathematician only after being published, even if 90 percent of published math is pretty useless and really shouldn't be what they are judged by.

I think if you do math and try to find interesting things in math, whether they were already solved or not, you're a mathematician. If you are a "published" mathematician, that might be a different title (like surgeon within the doctor field). Also, a "Ph.D. in Mathematics" is yet another title that may carry more prestige. I don't think that most Ph.D. mathematicians will think that way, though, because it's an inherent part of humanity to want to be better than people in some concrete way.