Is it correct to call oneself "Doctor/Scientist/Mathemathician" simply because one has a PhD in that field? If I use the word "Doctor" amongst laypeople, they will think I actually work as a doctor, treating people or advancing medicinal research on a daily basis.

If I use the word "Mathematician", they will think I work at a university where I spend some of my time lecturing students and some of my time doing research into complicated math problems.

If I use the word "Scientist", they will think I work in some laboratory where I am pivotal in the advancement of some cutting-edge research that will make the world a better place.

However, one can denote oneself using above terms by merely graduating and/or acquiring a PhD, even if one then left the field entirely, either out of choice or due to a failure to succeed.

My question is thus, is it wrong to call oneself a Doctor/Mathematician/Scientist if one is actually not working as one such, but merely acquired the appropiate degrees? Since clearly one is knowingly deceiving others*? If this question is too opinionated, I shall then ask: how many academics do actually use their degrees and doctorates to label themselves even if they actually are not working in those fields?



*To provide an example of such deceit, I had thought for a long time that Sam Harris, who writes many pieces on political, religious and scientific affairs, actually was a neuroscientist, as he claims. That sounded massively impressive to me the first time I heard it,  and added much merit to anything he had to say about science. I recently however found out that all he has done as a "neuroscientist" is get a PhD in it, and his thesis has been heavily criticized.
It's not the individual word that matters. Calling oneself a scientist is not unethical, no more than it is unethical to call oneself a musician without being in the Top100 charts. What is unethical is deliberately representing yourself in a way that causes people to assign you an authority that you do not have.

This is an incredibly gray area because it is entirely dependent on context. As an example, answering "I'm a doctor" to the question "so, what do you do?" would be misrepresenting yourself if you are in fact a mathematics PhD who now works in insurance. However, answering "I'm a doctor" to the question "how well-educated are you?" is perfectly reasonable.

The difference in understanding of a topic between someone who left a field a few years ago and someone who is currently still researching in it is probably not too huge, when it comes to science popularisation efforts. I'd also take issue with your characterization of "just" a PhD - PhDs are the frontline researchers of science, the further along you get in your career the less time you have to actually put your hands on the science. Professor is mostly a management job.

That said, you shouldn't be so impressed with someone that you defer uncritically to their opinion based on just a title. Every field has its marginal cranks, even actively currently practicing ones.