School counselors or guidance counselors are part of the history of professional counselors, but now are a different degree and license. Usually it is master's level. School psychologists can be master's level, and focus more on testing, and identifying students with learning disabilities, developmental disabilities, etc. Neither of these would usually do private counseling, unless they have a dual degree and dual license. If one has a degree in school counseling, it is possible to respecialize in mental health counseling with about a year of master's level coursework and an internship in mental health counseling. One can also respecialize from mental health counseling to school counseling, which also would require about a year of master's level coursework and an internship in school counseling. I know people who have done both respecializations - usually because they discover that they prefer the other work, but it would be possible to do both.
If you want to be a psychologist, private schools such as Argosy would readily accept you with either master's degree, but that wouldn't be most people's first choice, as these schools are very expensive, usually unfunded, and can be a challenge to get desired internship placements.
Another option for you would be to find a PhD or PsyD program that allows a combined focus on school psychology and counseling psychology. There are a few such programs at state universities, and are usually much less expensive, better funded, and have better statistics for internships and future employability. They accept students with a bachelor's degree or with a master's degree and may or may not allow students with a master's degree to count some of their graduate credits, usually not for more than one year of the doctoral program. Getting a master's degree first is not much of an advantage.
I have a master's degree in mental health counseling, and would like to become a psychologist, and have found that while I can work as a master's level counselor, which is better than I could do with my bachelor's degree, it is not an advantage for getting into a doctoral program (other than Argosy or other expensive private or online programs, which are not really what I want because the degree might not even be licenseable in my state, besides the huge debt).