First 2013 Bowman Aaron Judge Card?
 

Does anyone know which was the first Bowman Aaron Judge card to come out (2013 Bowman Draft/Chrome Draft #BDPP19, or the 2013 Bowman Draft Day Judge/Clarkin #DD-JC)? I don't think he has a card that actually says "1st Bowman", which would normally be the Bowman Draft. Or was there another Bowman to come out before these?

Bowman Chrome Mini #311 predated the draft set, I believe. The dual with clarkin was an insert set, so it doesn't really count. 2013 Sterling may have come out before draft as well.

https://www.comc.com/Cards,sc,=Aaron+judge,ot,i100

Release dates
Bowman Draft: November 13, 2013
Bowman Sterling: December 20th, 2013
Chrome Mini: Unknown, probably after Draft. Sold as factory set only.

Those both came out late in the year. Any idea when in 2013 the Bowman Judge/Clarkin Draft Day card came out?

It was an insert in Bowman Draft.

For that one year, Bowman stopped putting the 1st Bowman designation on cards. I don’t know why - it certainly created a lot of confusion and featured one of this generations most important cards. Huge mistake. His Draft is widely accepted as his first card. There’s no reason to worry about what day a Card was released - just the one that has held up best over time. Usually it is the players card with the RC logo. I think collectors drifted back to Judges Draft Card because he had so many 2017 cards that it became confusing. So you can’t go wrong with 2013 Draft - and not the one with another player on it.

The most extreme example one can cite to back up your statement is Mickey Mantle. His 1951 Bowman card came out before his 1952 Topps, but... I need say no more.