I work in a company that runs an own cloud for most of it’s business operations and for customers. I know where the data center is and when I go there I SEE the computers running the cloud.
It’s physical hardware running virtual machines and storage servers, and network switches with absurdly and unnecessary complex configuration, all owned by, well, someone else (the company).
So yes, the features of “the cloud” are distinct from your everyday stuff done on the computer sitting under your desk, but it really is just someone else’s computer running “the cloud”.