This was discussed before(cant remember where), but the people we meet in the "first contact" missions have obviously ALREADY been contacted by the Federation since they know about Federation history and culture. Diplomacy sure, by why misrepresent the missions with the name "first contact"? When people hear that they think they are going to actually MAKE first contact with an alien race.
I think the race hadn't met the Federation UNTIL your Boffs started briefing them (remember, the first contact is underway when you arrive in the Observation Lounge). They're trying to figure out what all the information your boffs shared meant
It seems modeled primarily on the "First Contact" reception from Insurrection, which seems to be the ceremonial name for the opening of diplomatic relations.
I agree, as a Starfleet diplomat you're sent there to make sure formal relations go as planned. That way you don't have some idiot phaser jocky telling a peaceful people that all he's done the past year is shoot Klingons, while ignoring the rest of the Federation and non-combat roles of Starfleet.