What would you say is the easiest way of extruding the two faces at the first screen so that the model ends up looking like the second?
In these situations I like to delete the edge and use F2 add-on for reconstructing the geometry(F2 comes with Blender and just needs to be activated in the Preferences). I have it enabled anyway for modeling and this process is quite pleasant because of the simplicity of it - only pressing f a few times:
This is using Blender 2.8 under Windows: Select the 'Edge' option in the Snap dropdown, but don't enable snapping. Select the vertical back face as shown. Place the mouse cursor over the edge indicated then press 'E' 'X'. Press the CTRL key which will extrude the face to the front edge and click the left mouse button. Select all with the 'A' key, CTRL 'M' to bring up the Merge menu and merge By Distance. Sounds long-winded, but only takes a few seconds.