In the vinyl release of the OST and the Art Book, this area is referred to as the Castle. It's incredibly clear that "Castle" is the area's official name. Besides, it makes more sense for New Home to be the specific city inside the Castle, like how Home is the city inisde the Ruins. We should at least mention this fact on the New Home page (although I'd argue renaming the page would be the most "canon" way to go).
The issue is that while Ruins is a full-fledged area, the playable gray area is just a few hallways, Asgore's house, and the throne room. I'm sure someone else can explain this better, but I think the house+throne area is typically called "Asgore's castle" (not a proper noun, just "castle") and while that's practically the same for gameplay purposes, as well as for the external references you mention, it does not mean New Home "is the Castle" or is "within the Castle". Nobody ever talks about New Home that way. New Home is a city that is vaguely near the king's castle, also known as Asgore's house.
I don't think we can really deliniate whether New Home is situated inside a castle or if the castle is situated inside New Home, or if the castle is New Home. All the save points say New Home so I'd be more willing to just use that and list Asgore's house, throne room, and the judgment hall as seperate rooms.
We also don't know if Home is a city inside the Ruins or if the Ruins are the ruins of Home. It's very nebulous.
The border for this area on console is called “Castle”. Can’t get much more clear than that