I looked at the netlogon service on the desktop and it was already set to manual. On the notebook though, it was set to automatic - so I changed that and restarted both computers. The pesky little box still comes up though!
I thought I had posted a screenshot of the box in question previously, although it seems that I took the screenshot but never posted it! Anyway, am doing so now...... and you will see from it that it definitely thinks that there is a domain involved somewhere along the line!
Btw, if you were tired there was no need to answer. I never mind waiting, even a few days would be fine. You Geeks are entitled to a life away from here!