"And if you're the 23rd reader, there is a 50% possibility that it is your bday, so happy birthday :)"

Oops! And you were doing so well. The probability that today (or rather yesterday when you posted) is the 23rd reader's birthday is 1/365, less than 3%. The probability that it is the birthday of one of the first 23 readers is still only 1 - (364/365)^23 , less than 7%.

As you showed above, it is only when each pair of birthdays are compared that the probability of a match reaches 50% for 23 people.