To think this is rude

Such bad manners, I agree with Maw & Buffybee nothing else to add.

I'd drop that " friend " like a hot brick, after telling her what I thought about their " private joke ".

You are definately not being unreasonable Oldwoman, they were very rude and mean, which makes me think that this woman is not the "friend" that you thought she was.
I am in agreement with MawBroon and would have just said, "Right, I'm off, I'll leave you two to it".
If you still want to go over to see her, tell her that you would rather it be just the two of you as you were uncomfortable last time when her cousin was there and couldn't join in their conversation.
Just tell her, what have you got to lose.

If it just happened once, it could have been excused as, indeed, a long-running family joke that would take too much to explain. However, to continue to make you feel uncomfortable and to exclude you is actually more unkind than rude. I'm sorry that your friendship is suffering and hope your friend will try to visit you for a change, perhaps to restore the balance.

I would have walked out. End of.
Dont say you are busy, say how rude you found her behaviour,
You have nothing to lose.

It IS rude! A bit like breaking into a foreign language that the third party doesn't speak.

I agree, rude and very bad manners indeed.