In the eyes of Microsoft, yes. They consider any printer that can be "seen" with various protocols to be a "network printer" in their "add a printer" wizard to be a network printer. Actually, that's not exactly true. They muddle it even more by offering only two types of printers, a "local printer" and a "network printer". In the eyes of MS, a local printer is really a printer that is locally administered, which may include network printers. A "network printer" is a printer that can be seen on the network, but it not locally administered.
For example, when I use the wizard to add a JetDirect equiped HP LaserJet printer (an obvious network printer) to a Windoze client, I select "local printer" even though the "network printer" is a more obvious choice.
No. It makes the combination of a print server and a printer a network printer.
Well, strictly speaking, any printer contrivance, that can be "seen" on a wired or wireless network, is a network printer. Are referring to the marketing terminology? If so, networkable or some such merely means that it has an ethernet port or wireless client and that it has some protocol (NETBIOS, LPR/LPD, etc) that allows computers on the wired or wireless LAN to print. Strictly speaking, a printer without a communications device is not a network printer, although it can claim to be one if the option is available to add ethernet or wireless.