You're misquoting or misunderstanding. The comment to which you are replying said:

    The paradoxy bits of the problem are the things that
    people find confusing. "Only 23?!" or "So if there are
    23 people in a room there's a 50% chance that one of
    them will share a birthday with me?"

In that comment he is saying that people mis-understand the question, and assume that it means that once there are 23 people in the room, then there's a 50:50 chance they will share a birthday with them specifically.

And that's exactly the wrong question, as you point out. So when you say:

    Is "So if there are 23 people in a room there's a 50% chance
    that one of them will share a birthday with me?" correct?

No, that's not correct, but it is what people think they hear, and it's that confusion that makes this whole thing sometimes called a paradox.

So let's be clear:

    If you're in a room with 22 other people, the chance
    that one of them shares a birthday specifically with
    you is nowhere near 50%

    However, the chance that among the 23 people in the
    room there is, somewhere, a shared birthday, is indeed
    slightly greater than 50%

And my experience is that it really doesn't matter how carefully you word this, some people simply will not understand it.

You're right, I misunderstand the comment I replied was using that as an example of how other people misunderstand the problem. D'oh. Edited!