But the characters are connecting the adventure sites together. Or are you trying to say that a sandbox campaign can't be a "campaign"?
No of course not.
But, since the adventures don't actually share characters, how can they all be considered the same campaign?
None of the adventures have any linkage in your example, and none of them share characters. While some characters might share two or three of the five adventures, taken as a whole, none of the adventures are actually linked together in any way.
How is this not simply a collection of pick up games? The only connecting element between any of these is the DM. It's a campaign if all adventures share a DM?
Well, no, that doesn't work because lots of groups rotate DM's.
I do think Hobo has the right of it here. Campaign is best defined by its center, not by its edges. You will never get an air tight definition that will satisfy all examples perfectly. It's just like the word "forest". What is a forest? At what point does a collection of trees become a forest?
But, if I'm standing in the middle of one, I know it's a forest.
For me, the example you gave, with no further information, I would not call a campaign. It's a group of one shots. No one in the group actually shares experiences beyond a couple of the adventures.
Now, if there was something linking all the characters together, something linking the adventures together (other than simple geography), then I'd have no problems calling it a campaign.
But your specific example? No, I wouldn't call it a campaign.
Look at the Westeros campaign. There is a common linkage between all the adventures - exploration of the West lands. Every adventure is meant to expand the knowledge of the group of the unexplored lands. And all the adventurers are linked together by being members of the same adventurers guild - with shared information and shared goals.
That's enough linkage for me.