As others have stated below, in this context you have bare metal and you have the cloud. You could also add in virtul machines, which run either on premise or in the cloud.
Thousands of physical servers our company manages that aren't rented, but owned by us. Does that help?
Yes. Thank you! If only industry marketing hadn't coined the terms, we might have: cloud -> provisioned server, bare-metal -> managed server or some other less context dependent terms.
Just for the curiosity, "managed server" in this area is already taken, and means that you get a sysadmin with the rented server (the server is managed for you, including installation and maintenance of the software you need on it). It is "higher level" than cloud servers, not "lower level" ;)
Thousands of physical servers our company manages that aren't rented, but owned by us. Does that help?