I'm sure it was meant in more of a "dual-citizenship" way, I can only imagine how complicated the logistics of being part of two interstellar species would be.
Edit: I just looked at it on a larger screen. It appears to be an issue with using a small font size. The i's do have tittles they've just blended together on the smaller font.
There are two TOS episodes that directly referenced the "United Earth Space Probe Agency" as the operational command for the Enterprise.
Basically, the United Federation of Planets did not exist when the show started, and its basic structure wasn't settled on until well into TNG. There's a second season TNG where the Klingons are federation members.