Definitely the vast majority of parents/people do not understand what we do. I've been called the "district tester" and then other people believe I do mental health therapy all day long (which is fine, I'm trained to do that, but I have no time to do that!). Consistent with NASP's ideal service model, I was trained for counseling/ therapy, assessment, and consultation. But yes, like you, most people who don't know seem to think I do therapy all day. Definitely a lack of understanding from others of what I do.
Educational Psychologist would still have the word "psychologist" in it, so I would think they would still think of you more in the clinical sense. I prefer to call myself an educational detective, so I think that would be the best title
I'm really interested in knowing what other school employees think of our job titles!