In all sorts of job situations we have official ranks. I held the titled position of Third Horn in the Nashville Symphony for six seasons for example. Later I was hired at Arizona State University at the Assistant Professor rank and currently  I am an Associate Professor that just finished my tenth year on the faculty. In the future I certainly aspire to achieve the rank and salary of full Professor. [UPDATE: I am now, as of 2018, a full professor at Arizona State]
All horn players should strive for 100% accuracy. In my bio and publications I opt for the 100% accurate title of Associate Professor. The term Associate Professor will always confuse some readers outside academia, it sounds like I am some sort of associate to a professor, but no, I am the horn professor at ASU and that is my sole job.
Assistant Professor, Associate Professor, and Professor are all full time positions; I do still perform but the thrust of all I do is toward my full time teaching. Typically, college professors in full time positions are hired at the Assistant Professor rank, and progress with tenure to Associate Professor and then later on to Professor.
Some teachers out there don’t use completely accurate titles in their published bios or other materials that come out in ways that border on deceptive. Students looking at schools or those mentoring people looking to study the horn need to be very aware of this as you can be fooled.
There are different tactics that horn teachers can use toward this issue. One option that is honest is to say that you are “horn professor.” This is accurate technically; you are not saying you have achieved the rank and salary of full Professor, just that you are a professor, of some sort, at that school. But I am sure there are people out there who call themselves “Professor of Horn” when in fact they certainly are not, and perhaps are only even in an adjunct faculty line in reality. The word professor in their case does not occur anywhere in their official job title.
Professor, in the context of teaching in the United States anyway, is not just an honorary title to throw around; it means something very specific in an academic setting.
If you are in doubt about if someone is really a professor or not check the official website of the school in question as the accurate job title should be posted somewhere. My official title presently (2011) is actually Associate Professor of Music, which is very typical among schools of music.
In relation to this topic the worst case scenario would be in a job search; the resume or vitae must be completely accurate as to the use of these terms. If not, it will cause problems in the end. Never fudge stuff like this on a resume; 100% accuracy as to job titles is the right way to do things.