Open Norton and go to Help - About and at the top of page it will show Device name. This should match the My Device page. You can change the name on My Device page and it will update the Help - About Device name.
The My Services page shows the Windows PC name.
Jim 
liquidmetal
What you are seeing is, unfortunately, a common problem. You can find many posts here about the Devices page not matching the Services page. The Services page is usually correct. So in your last image, it shows the 4 devices are activated and protected.
What you can try on the PC's is going to the main NSD screen and clicking on Help - Subscription Status. This syncs your PC with the Norton subscription servers.
@20750065, thanks for your reply and suggestions.
Remote management option is set to ON in Norton Security Deluxe on the Windows 10 desktop
I went through the KB article and couple of the specific items namely "I see an invalid product key error" and "My Norton subscription shows as expired" though they don't really apply to my situation.
Also checked the other page to determine status of protection, and on the desktop, as stated previously, it does show "You are Protected". I also tried the 'Renew' button on the NSD, clicked the "I have a product key" link and entered the key as shown on the myaccount webpage. Here I got the message that "it has already been activated" (-1016).
So I'm not really sure what is going on. Looks like I'd have to uninstall NSD from both the desktop (Windows 10) and laptop (Vista) and try again?
@liquidmetal,
Plz ensure you've input correct machine info. And to re-check yr Norton subscription status, plz read this KB.
And, plz ensure you've enabled the option "Remote Management" under Administrative is ON.
Then, pay a visit to this page and make sure your devices are protected.
Thx and luck~ :)