Generally speaking you aren't renting server space, you're renting server equipment i.e. the rack, hardware, power, networking, so I would imagine that would or at least could be the difference.
I read the article, they do say "colocation"; besides which, XO Communications does not rent servers as far as I know. So that seems to point that they were renting the physical space that their servers are in.
Would it make a difference if they were just renting a partial rack, or a full rack that goes all the way down to the floor and is enclosed? I don't know ...
The contract I signed, had a clause that said if they had to move me somewhere else, they would pay all costs of doing so. I don't know how standard a clause this is.
I rent a furnished appartment, and while the company that owns it would never touch it without my permission, the contract is very specific about it being this building, they would never say "mind moving next door?" whereas with a rack in a data centre, not only does the contract not specify exactly where the rack is placed, it makes very little difference about it's exact location.