Phil Spencer has refers to first party as games he publishes including partner games they don't own (Sunset Overdrive, Alan Wake, Ryse) so when he says "1st party" - and I've seen this mentioned in a few threads, he is referring to anything they have published directly with the developer, as the best definition I've got.
When MS say they have big 1st party presence at E3, I don't think he is specifically referring to internal studios (although they will be showing stuff) and people should probably set their expectations accordingly for that.
2nd party is stuff like Dead Rising 3 (and possibly Tomb Raider) which he published from a third party. Not sure where Titanfall stands in all this, EA published it so I guess its just a briefcase of money exclusive.
My personal definition is who owns it. Microsoft own Ori and State of Decay, its on the platforms they allow because they own it and won't be on competitors platforms, so its 1st party even though Moon and Undead Labs aren't owned by MS.
2nd party, Insomniac/SO, Crytek/Ryse, MS published it but don't own the game or studio.
3rd party, where they paid a publisher for exclusivity whether its full or timed, whatever.