We use Ggeneral (neutral) introductions for people we might never meet again, for meetings which are not very important, or for meeting people like ourselves, for instance the people we will work with.

· Jordan: "Excuse me, I know you by sight from the gym, but I don’t believe we’ve met.

Language wise, this is all correct, but "I know you by sight from the gym" is kind of a mouthful for a first introduction, so I don't think anyone would actually start their greeting this way. If you're in the gym giving the greeting, you might say something like "Hey, I'm Jordan, I see you here a lot"

It’s nice to meet you officially."

"meet you officially" sounds like you're meeting them in the course of some official duties, like a diplomat who has talked to you in the bar, but now you're talking official business before.

Please tell me if it's okay to say, "It's nice to finally put a name to the face"

I think it's fine. I hear the inverse "It's nice to put a face to the name" much more when it's e.g. someone you've talked to over email that you're meeting in person for the first time, but I think this way works too.