I’m looking for the name of the map projection that is essentially a picture of a globe- is this Azimuthal, or is this just a general category?

Azimuthal is just a general category.

What you are looking for is probably General Perspective projection, which is a type of Azimuthal projection.

Wikipedia article on map projections in general:

List of azimuthal projections (and other types as well):

General Perspective projection:

Thanks! Much appreciated.