I think it would be unlikely for one of my peers (mostly developers) to mention "The Cloud" unless they were trying to be intensionally buzzwordy for comedy effect. They would be more specific.
I/We would however say:
a cloud to mean clusters machines (probably split into VPSs)
xxxx's cloud to mean xxxx's server infrastructure/platform
I like The Cloud as a common term to mean "Stored/Processed out there in the internet somewhere", not as a well defined technical term.