A computer connected to the network is called?

Speaking as a network engineer of 25 years' experience, most often the computers that are connected to the network are called stations or workstations. Those that are devoted to storing data, providing print services, etc. are referred to as servers. Most often nowadays, we do not have dedicated servers as were most common in years past. Since so many now use Windows as the choice for their server, they are frequently set up in a peer-to-peer network. Only if someone sets a machine aside and actually installs some version of Windows Server on the machine is it set up more to be a server but still, technically, it is still in a peer-to-peer situation since the machine can still be used as a workstation. A dedicated server cannot be so used, such as the case of the Novell Netware servers that were common until about a decade ago.

When one speaks of clients in regard to computers, that is generally in regard to licensing (as would the term "seats") but the engineer would most often not use those terms nor would the help desk.

As for the term node, that is a term that a network engineer or technician would use when referring to the logical design of the network but once again this would not be used when speaking with a network user. To put it simply, essentially any device that can accept an IP address is a node. As such, managed switches are referred to as nodes but unmanaged switches generally are not.