It's been a minute since I was an active ArcGIS user, so please excuse me if my understanding is a bit outdated.
From what I know, accessing database layers in ArcGIS desktop requires importing data from a server. The ArcGIS app does not synch tile layers to a server, but rather creates a link to the tile layer that can be input into a certain project.
So, if you wanted to utilize your maps in the desktop version of ArcGIS, I think the exported GeoTIFFs or DEM files are the simplest options. This should also ensure you are using the correct projection system. The web tile app is set to Web Mercator.
Anyone who works with ArcGIS more regularly- please correct me if you know otherwise! Thanks in advance.