Webster's dictionary defines it as "a specialist or expert in mathematics."
There are several hundred areas of mathematics where one could spend the rest of their
life doing research in mathematics. PhD's in math typically know MUCH LESS than
1% of the mathematics known worldwide today. I'd say that anyone who is willing to call
themselves a mathematician probably is one.
Now if you are talking about a "working" statistician, they go through a battery of tests that
would make diamond wilt. It's not an easy road, but usually quite lucrative. They are
mathematicians, but usually only expert in a small area of mathematics.
And today some of the most active mathematicians use computers to bounce their ideas
off of often. Wonderful tools for exploring patterns, testing formulas, etc.
And age doesn't necessarily have a lot to do with it. Gauss could have probably been called
a mathematician before he was a teenager. He taught himself to add columns of numbers
by the age two and a half. I was still trying to get my walking down about that age.
Have a super day!