When we (locally) talk about having your own hardware we use terms like 'bare metal' and 'machines' or even the specific names of the types of machines, like 'the blades' or 'the dells'. We really only use 'datacenter' when we talk about a physical location we own, with power we own, cooling we own, networking we own, and access control we own. Otherwise it's just colo.
I live and work in the Bay area. I was part of the larger org that ran Uber’s datacenter (co-located) and that’s how it was talked about.
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.
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).
If someone says "I have a brownstone in Manhattan" and they actually mean they rent one of seven rooms, it's just a lie.