The Esri default print service provides very basic printing. It is not intended to produce high fidelity, large format printouts. You are correct when you say that the printing options vary somewhat within the platform. Currently, the most configurable printing can be had from an application created from the Web App Builder.
If your organization uses the default print service, layers that are not accessible externally, time-aware layers, and KML ground overlays and network links without refresh properties do not appear on a printed map. If your organization uses a custom print service, certain types of layers may not appear on a printed map.
An organization can specify their own print service, that is running in house on an ArcGIS Server instance that is externally accessible. This allows you to create your own layouts and expose them in your custom applications.
Hope this helps,
Mike