Why would you call colocation "building your own data center"? You could call it "colocation" or "renting space in a data center". What are you building? You're racking. Can you say what you mean?
Come to my office and tell me how it’s not actually my office because it’s leased by my company from the investment vehicle for institutional investors that owns the building that stands on land owned by someone else again that was stolen by the British anyway and therefore calling it “my office” makes me a fool and a liar and I should just “say what I mean”.
When you invite a girl/guy over, do you say "let's meet at my place" or "let's meet at the place I'm renting"? The possessive pronoun does not necessarily express ownership, it can just as well express occupancy.
I wouldn't oppose telling a client "we can meet at your data centre". I would not tell my wife "we need to discuss building our apartment complex" when we are planning interior decorations in our flat.
If I said to my wife, “let’s build a home together”, she would be halfway done with engaging a promising firm of radical young architects and negotiating for April delivery of pre-stressed concrete, Italian art glass, and Japanese tatami mats by close of business.