I challenge your tagline:

While this statement can apply to many things, in the case of language (or currency value, or rules of etiquette, or what day a holiday is celebrated, or any number of things), the truth ABSOLUTELY changes because of what people believe.

Words have meaning because people decide they had that meaning. Presently, the definition of "mathematician" as defined in a dictionary or by common use, is not as specific as you propose. People and dictionaries define the word to mean a person who specializes in mathematics, or in even more casual vernacular, a person who is skilled in mathematics.

You can wish this to not be the case, but that doesn't change reality. If mathematicians (that is, the kind you consider to be "real" mathematicians) care, a more realistic and constructive solution might be to self-assign a more specific designation, in the same way that theoretical physicists do (in order to differentiate themselves from, for example, applied physicists, who often do not have PhDs or do work that advances the science of physics, but are nonetheless valuable physics specialists who do very important work and should rightfully, IMO, call themselves physicists.)