Perhaps you haven’t noticed this, but there are only 12 months in a year, not 24.
The date style used on the sites themselves (Sep 13) seems to suit everyone, so why not go with that? Personally, I’d prefer the date before the month name, rather than after, as this better suits the way I speak dates: I say the third of October or the twenty-fourth of the fourth, not March fifth. However, the way dates are written here is, at least, not confusing.
The problem with dates is that they have written representations, and they are also spoken, but the two aren’t necessarily intertwined. In other words, when saying a date aloud, I don’t generally read it, I simply say it.
J.R., English Language & Usage.