Good question. Probably the same. English, and especially business English, is not a precise language. I am not really an expert but was part of the larger org.
Just to be clear - colocation here is in order of thousands of square feet. The datacenter provider provides redundant utilities. The customer does everything else.
Seems a little silly to blame english when people bend terminology to muddy the waters on purpose. When 'serverless' means easily using a server and 'we have our own datacenter' means 'we don't have our own data center and rent from someone else' maybe the problem is people trying to stretch the significance of what they are doing.
If someone says 'I own my house' and they actually mean 'I rent an apartment', no one is going to say 'oh well, english is an imprecise language'.
In all the examples given earlier, the words "I have a datacentre" (or the approximate "I have my own datacenter") vs "I own a datacentre" (or the approximate "I own my own datacentre") have very significant differences.
In the UK, many people would say "I have a flat in this district" and that would be understood as being a renter, since some UK neighborhoods are just too expensive, and renting is the norm rather than ownership.
Similar in North American cities "I have an apartment in a brownstone in Manhattan" - does not necessarily mean they own it, but it could mean ownership or rental or lease or rent-controlled/stabilized tenancy (which could go on for multiple decades/generations of inhabitancy).
Just to be clear - colocation here is in order of thousands of square feet. The datacenter provider provides redundant utilities. The customer does everything else.