Oy. The Forgotten Realms is one, single "setting." Do all of the adventures ever played in that setting constitute one, single, unequivocal "campaign?"
No.
BUT, do all the adventures ever played in that setting
under the same DM represent one campaign? Probably. I'd leave that decision up to each particular DM.
Sometimes, a single campaign can run under several different DMs at different times, though this would be tough to pull off well.
As I understand it, thousands (I'm pulling this number out of a hat) of different DMs have created their own, unique campaigns set in the Realms; and the outcomes for one DM's campaign might well (and probably do) contradict the outcomes of another DM's campaign in the same setting -- yet they are all "adventures in a setting," so must be the same "campaign" if your second equivalency is correct.
You need to factor in the DM.
For my own games, I define the campaign by setting provided there's any interaction between the various parties. My current setting/campaign has no less than 4 active parties* in the field, each with characters who have already adventured with characters in at least 2 of the others. And though those parties are currently playing out 4 different and mostly-distinct story arcs, as far as I (and the players) are concerned it's all one campaign.
* - lest you think I'm running 4 games a week and thus have *no* life at all, I run two of the parties each week with the other two on hold; and occasionally we switch 'em up. This also means I can gleefully TPK at least one if I want, because there's backups waiting in the wings!
That said, before this one myself and another DM ran what we consider to be 5 major campaigns. He has managed to interweave his three such that there's a growing case to be made that they are in fact all the same great big one (which, as an aside, would now be in its 29th year if true). A lesser case could be made that my previous two are also involved, given the amount of interaction between my campaigns and his over time. We have, however, thus far by choice kept them as five distinct campaigns.
Lan-"then again, maybe I don't have a life"-efan