Do the terms “client” and “server” refer to computers or programs?
Both, it depends on the context.
A server program is usually called a “daemon” in Unix-like operating systems, unless the program has a specific name already. Note that a daemon can be a client, as well.
There is no specific name for client programs.
It is best not to classify computers as servers or clients. Instead, it is better to indicate what kind of server and/or client programs run on a specific computer. Alternatively, one can also indicate what services are offered by a computer.