HI,
Not sure what core means
You showed where it says 10 pro but not a screen shot of where it says home version.
An upgrade from home to pro is discounted pretty much in half
Pro is 200.us +.. so an upgrade from 10 home is approximately 100.us + might be 120.us if I remember correctly I did actually do an upgrade at the MS store.
Sorry about the delay. Here's a screen of Control Panel/System:
What's shown under Devices online in your Microsoft Account can be outdated and inaccurate. I regularly have to clean out dead wood. If I do a clean install on a Device Windows insists on giving it a weird Device Name. I then rename it to what I want and get a second device listed with the new Device name. It's getting better, it does seem to eventually update all on its own and remove the junk. Delete the one that's there and a new accurate listing should show up. Assuming your logged in with that account. Or just wait and see if it gets in sync all on its own.
Not sure on the home vs. pro issue, but I do see one major item you need to correct in some fashion: You have 8 gigs of RAM, but you are using a 32 bit version of the OS. You are not able to use 1/2 of your available RAM because of that - 32 bit OS can only address up to 4 gig of RAM. Did you add RAM at some point after the OS was installed? No other way around that except a full reinstall...
Also wanted to add that the only potentially free upgrade to Windows 10 pro is (I think) on the Surface Laptop running Windows 10S:
Microsoft extends free Windows 10 S to Win 10 Pro upgrade offer • The Register
"The OS comes with restrictions: you can only load apps from Microsoft's own Store and Edge is the default browser. Even if Microsoft's store admits other browsers, Edge or IE will remain the default apps for opening .htm files. Bing is the default search engine in Edge and IE and there's no option to replace it. Microsoft justified those restrictions by saying they help keep Windows 10 S easy to administer and use. But it also said users who can't get the apps they really, really need in its store could upgrade to Windows 10 Pro, for US49. Buyers of Microsoft's own Surface Laptop, a device at the higher-end of 10 S devices, were given until December 31st, 2017, to upgrade gratis."
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