This is probably not the definitive answer, but usually my customers want the string action to be as low as possible without buzzing or rattling. Some professionals, however, ask for a higher action because it suits their style of playing. Perhaps this is what is meant by the term "professional" height.
Karl Roy, who wrote a definitive book on string instruments, gives the height of the E string at the end of the fingerboard as 11 mm. This is fine for a synthetic or gut string. In my shop, the height comes down to 9 mm for a steel E as a good average number, although individual preferences can nudge this even a little lower.
