Originally Posted by Michael J Fox is Canadian
It seems at though 'old' doc was always the one who got arrested for the speak easy in all the game versions of the timeline with the exception of the burnt hill valley one; in episode 3 we see Old Doc in FCB's picture, and in episodes 4 and 5, everyone remembers 'Carl Sagan' in 1931 even with FCB being present there. Here's how each of the docs timelines should go linearly;
Goes through the time travels in the trilogy, eventually settles with Clara in a time zone other than the 1980's, goes back to 1931, gets arrested, goes back to 1986 briefly, and back to 1931 before being erased due to the FCB timeline
lives naturally, with his only time travels being back from 1986 to 1931 and several in 1931 before being erased
seems to have been through all the time travels in the trilogy as he has the hoverboard and has been with Clara but the main difference from old doc is he has lived the past 7 months in the 1980's between the trilogy and game. He does seem to remember his original visit to 1931 including his motive.
I think you guys overcomplicate this a bit. The "new" Doc is basically the same as before, the only difference is that thanks to Marty he's able to come to terms with his father. Because of this he becomes more attached to his father's legacy and Hill Valley so he doesn't leave his home at the end of the third film. Apart from this his timeline stays the same as before because there's no interference to it since 1931. He goes to 1931 for the same reasons as the Doc in the beginning of the game and everything happens to him the same way because there is nothing that would change that.