Thanks Saab - that makes sense.
For SoCal - I have obviously confused you for which I apologise. By 'Desktop' I mean the screen with all the Icons. I have now discovered (via the laptop) that the (correctly functioning) 'Safe with Networking' desktop is identical to the 'Safe' mode desktop - no clues that I am in 'networking' which is one less question. As for the Boot Menu, yes that is correct and yes, I am selecting the correct line.
Regarding 'comparing' the two machines - I am stuck - there appears to be an inscrutable problem to which no-one has a solution at this time, so I need to establish some baselines. I will look at Services next on both - is there a service that needs to run in order for the ethernet adapter to even 'appear' in Device manager? Is there a simple way to see which 'Safe' mode I am actually in?
Is there any point in trying to re-install the ethernet drivers in 'Safe' mode, when they are already present and functioning in 'Normal'?