Karate = dojos surely? If it's karate there should be details on the style etc, if it's TKD or TSD then why are they calling it karate? Confusing!
It is pretty much the norm in the States for taekwondo schools to have 'Karate' on the sign. This goes back to the seventies when nobody knew what taekwondo was, but most everyone had at least heard of karate.
So far as I know, and recall from my tenure with TSD, TSD is Shotokan with Korean terminology. They use the same forms (so far as I know, the Pyung-ahn forms are the same as the Pinan and Heian forms). Calling TSD karate does make sense.
I am of the opinion that taekwondo falls under the general category of karate, as it is Shotokan derived and has more in common with Shotokan than it does with anything that is native to Korea.
But knowing whether it is karate, taekwondo, or TSD is fairly important, as it determines the next set of questions: type of sparring, parent org, and which style thereof.
If it is karate, Japanese or Okinawan? What Ryu?
If TKD, Chang Hon, Kukkiwon, or something else?
If TSD, I am sure that there are questions as well, but I am not familiar enough with the permutations of TSD to know what one would even ask, let alone what the answers would be.
Certainly, a karate school with Korean writing in the windows and on the belts is a good indication that you are not dealing with authentic 'karate' but, with something Korean.
All things considered, calling TKD or TSD karate is not entirely dishonest, but I do believe that the practice has outlived its usefulness. Simply putting 'martial arts' on the sign would suffice.
Daniel