You have to pick:

I'm assuming this is the American president.

I think it's supposed to be the president (or equivilant thereof) of the country you live in.

President of what?

Would I gain a power or would I just be a member and attending all the luncheons?

So I could be a family friend of a political hack? No, thanks.

Or, perhaps, become one of the existing members (eg Wolverine, Cyclops, etc.)

You'd have a power (you can't be a member otherwise). The power would be on par with a typical x-man.