Does it mean the server is part of the maker?
can be thought of as the piece of equipment used to make or brew the coffee.
In your example, the
is the vessel used to deliver the brewed coffee especially of the coffee is brewed in a large urn. On many occassions, the coffee is brewed into a vessel, the coffee pot which can then be used for serving (your Mr Coffee is an example).
No. Coffee makers usually include a coffee server when you purchase them. They just complete each other and sometimes they are sold separately.