Learn English – Usage of “you any good?”

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.

Now here...

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.

Something like...

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.

To know someone is to be familiar with them, or to know what kind of a person they are.

You can know someone or something better, meaning to be more familiar with them, or to have a greater knowledge of them.

When someone says "You should know me better than that", that is referring to how well you appear to know them, and they are saying that they expect you to be more familiar with them. In your example, you said that the man did not seem to like you, to which he replied that you should be more familiar with him than to think that he did not like you - that is, he does like you, and he would expect you to know that.

You use this kind of phrase when someone has expressed an opinion about you that is not only wrong, but that you are surprised they would have, because you expected them to be more familiar with your character. For example,

"You think I would cheat at a friendly game of cards? Come on - you know me better than that."

(that is, the speaker is expressing surprise and dismay that someone would think that he would cheat at cards)

or

"Do we have to bring potato salad to the Johnsons' picnic too? We're already bringing dessert." "You know me better than that...I can't bring just one dish!"

(that is, the second speaker is saying that the first speaker should know that the second speaker is not the kind of person who would bring only one dish when invited to a picnic)