How can turning right on a red light be considered running a red light when there is no sign prohibiting doing so?
In some states, you can make a right turn at a red light if you come to a complete stop, look to make sure it is safe to make the turn, and then make it. In other states, it is always illegal to make a turn on a red light, even if you are out in the desert at 3 a.m. at the only red light within 100 miles with no cars in sight.
In any state, if, when you made your turn on a red light, cars had to screech to a stop or pedestrians had to fly out of your way, you are pretty much out of luck.
The best way to determine whether the officer who gave you the ticket knew his or her stuff is to look up the law. Look up your state's vehicle code at Nolo's Legal Research Center or a library, and then find the statute number (most likely written on your ticket). Read it very carefully to see whether you actually violated the law. If you didn't, go to court with your ticket and a copy of the law and plead your case.