Li Shenron
Legend
My friend views a campaign as a storyline. He usually comes up with a lot of campaign ideas, and is fine if it lasts a year or so (considering that a long campaign) or even shorter if it is intended to be a shorter campaign. Because he is always a few campaign ideas ahead of what we are currently playing, he supports characters with very unique concepts that might only work well in certain scenarios. The campaign covers a short time period in the game world also, anything from a few days to a couple of years, and follow-up campaigns with the same characters are completely possible. In short, a campaign for him is like a good long movie trilogy or series, say Lord of the Rings or Harry Potter.
I am with your DM, except for the fact that I would not call it a campaign unless it lasts long enough also in fantasy world time.
But the definition is blurred for me too, because I don't know how to call a game where the campaign strictly speaking (i.e. the main story arc) is over, but the players decide to continue playing the same PCs for more adventures. Is it still the same campaign? If you start a new story arc it's maybe a follow-up campaign, but if there is no additional story arc but only small adventures?
Anyway overall I used to think the term is derived from the military language, which strongly suggests that a campaign ends with reaching (or failing) an objective:
http://en.wikipedia.org/wiki/Military_campaign