Dear Christine_Asus, thank for your kind reply.
Concerning your questions:
When I changed the name of my Google Account under Setting>Google the new name was shown. Under Accounts I am no more sure , I returned to the old name in the meantime: my impression is that the old name was still shown in the top title Accounts for 'name' (in the detail list of accounts the Gmail email address is shown for the Google account and the Outlook email address is shown for the Microsoft related accounts.)
In any case: now that I returned to the old name of my Google Account, the old name is shown both under Settings>Google and under Settings>Accounts ( top title Accounts for 'name' ).
Concerning the step of resetting the phone to factory reset, I am pretty sure that, as a result of it, the desired result will be achieved and the new name of the Google Account will also be shown under Settings>System>About phone. However it is not worth for me doing this at this time. The phone is working fine , I am using it extensively during the day for calls, emails and messages , I prefer to keep it available. As I said, having returned to the old name in the Google Account, everything is looking coherent inside the phone itself. Overall, the issue seems to be mainly of a cosmetic nature. It may well be a design feature. Thanks , in any case!