What was that first wedding?

This will have to wait until the Blu-ray to corroborate, but I think there was first a civil ceremony, then they had some problems, leading to the religious ceremony (possibly after she had left him and then came back?). And possibly there was also the intercurrent issue of whether she was divorced yet? But earlier than that there is a wedding which Quintana presides over, which has nothing to do with the main characters except being I think where we first actually meet Quintana. Sorry, best I can do at the moment; undoubtedly others remember better. 1John4:4

there was the wedding scene with Javier Barden and my understanding of that was just that Barden was depressed, there were happy people around him and he was in despair nonetheless.

Malick films can't be read literally. I took it as a sort of misdirection as you assume maybe Marina and Neil are getting married but thats not how it will go down for them.

Perhaps a way of reading it would be to show a normal happy wedding and then later show Marina and Neil at the courthouse where a criminal signs off as the witness to the marriage.

In reality though, not even Malick knows what this film is about.

My interpretation was that was Malick's formal introduction of the idea of "two kinds of love" - a Christian concept with one kind of love being the sexual kind and the other kind of love (usually associated by Christians with the Greek word "agape") being "selfless". What better way to have a scene illustrate both kinds of love and draw a sharp distinction between them than showing a celibate priest officiating at a marriage.