They aired "The Man Trap" first.
Wouldn't that make "The Man Trap" the pilot? Usually the pilot is the first episode.
The production pilot is generally the sample episode which sets up the premise, or shows the networks an example of the series format. These episodes may or may not be aired first, depending on the network and the series. If the entire premise is set up in the pilot, then it (or a re-edited version of it) is aired first in the run. However, if the episode can essentially be shown anywhere, a "stronger" episode may be aired first. This is called the "broadcast pilot." For example, the series "I Spy" began its run with "So Long Patrick Henry" (an amazing episode written by star Robert Culp). However, the series pilot was actually "Affair in T'Sien Cha" which was a bland entry aired 14th, between Christmas and New Years, a graveyard for new programming, and a relatively harmless place to stick crappy episodes.
The Cage and Where No Man Has Gone Before are series pilots.
The Man Trap would be considered the broadcast pilot, although I'd it "the premiere".