Maybe not. When Kirk and the landing party are first captured in "Bread and Circuses", they simply lie about where they're from and that the Enterprise is a ship at sea.
In that particular case it was easy for them to blend in with the rest of the apparently Caucasian population.
On the other hand, Tracey and landing party looked like civilized savages to the Kohms. Certainly he must have done some explaining, but what really made me scratch my hand is this "arranging a planetary survey".
Does that mean the more advanced the alien civilization is, the more important it is not to hint that you're from outer space?
Do they believe that their appearance in "The Omega Glory" will be dismissed by later generations as a fairy tale just as we dismiss dragons?
The Kohms recognized the Enterprise landing party's phasers as "fire boxes", obviously because they had seen Tracey use these (which is what makes Spock raise an eyebrow, IIRC).