I think I'm going to start sounding like a broken record, but this is another example of something that is "style" and not "grammar." I say "style" when I mean that there is no different in meaning or logic, and it doesn't violate any rule of grammar. When there are choices involved, you pick a style and stick with it.
First of all, in the US, the month comes first. But I think we're unique in that. My birthday is May 23 or 5/23 here, but 23 May just about everywhere else. (You are all now on notice - you have just under five weeks to prepare an appropriate celebration.)
Rarely is the rd or nd used in text. I wouldn't write that my birthday is May 23rd, and rarely would you use an abbreviation like Apr. in text. But it all depends on your house style. Chicago Manual and AP disagree on a few things.