English Word Request for a Certain Type Of House

It seems to be called an atrium house. See Best 20+ Atrium House Ideas on Pinterest for pictures of modern houses built around an atrium. If you were to hire an architect to design you such a house, you might start off by saying you wanted an atrium house.

If you have a big budget, you might want something like the Loma House. The Loma House is not called an atrium house in this article from Arch Daily, but is one of the 20+ Atrium Houses on Pinterest.

(I wanted to put a picture in this answer, but was unable to do so.)

Here's a photo of a house with an atrium, defined in the accompanying article as an open space surrounded by the house.

The Eichler houses became especially popular in the late 1950s (and on) in the San Francisco Bay Area, where they were relatively simply built; they are now much sought after, sort of a revival.