I searched through the documentation and examples but could not find a complete list of network types. I looked at the OSM API documentation as well. I know a few like "all", "drive", "bike", "walk" but is there a complete list somewhere?
See the documentation. There is a list of possible network_type arguments in the docstring of every graph_from_whatever function. To answer this for modern OSMnx versions (v2.0 and on), the built-in network types are: "all", "all_public", "bike", "drive", "drive_service", "walk".
"all" provides you with all private and public (including service) roads/paths.
all_public provides you with all public (including service) roads/paths.
More details in the docs and demonstrations in the examples gallery.