I may apply general logic here:
semifinal -not fully final;
semisolid -not fully solid;
semi first-name -not a full first name
I'm not an American so exactly don't know how to shorten names. But let me try...
For instance, your friend's name is Rachel Fishman.
Fishman: Last Name
Rachel: First name
Rach: Semi-first name.
Said that, if you are close to Rachel, you don't bother uttering the whole name. What I just said up there...
Hey Rach, bring me beer!
So, if Rachel says, "Let me introduce myself on a semi-first-name basis", I think Rachel is trying to be friendly with the audience/opposite person. Said that, on the very first meeting, Rachel wants to be an amicable person and does not want to get into formalities.
Hey, that's okay, call me Rach!
Immediately, the opposite person gets feeling of friendliness.
[In your case, as I said earlier, I'm not sure how to shorten the word 'John' further!]
This is a logical answer, and I'm not sure whether it has something to do with 'culture'. Depending upon comments, I shall improve/delete this. Here, I just tried to help the OP.