There’s no special name, just a special site. It’s basically like the new app, but in landscape mode on a gigantic tablet.
At the moment you can see all your devices, Rooms, Scenes, and SmartApps (mislabeled as Automations on the web app). Automations (with regards to the new app) are not there yet. The web UI is similar to the new app UI, but there are still a lot of differences.
Here’s a screenshot or 2 or 3:
Note that there’s a “People” group. Mobile presence devices (our phones) actually show up there even though the new app doesn’t support that yet. You’d think that if the ST guys can get phones to show in the web app and Classic, that doing this for the new app shouldn’t be too difficult.
I’d likely rarely use this because you can’t move around Rooms or Groups (like Scenes, STHM, etc), and since landscape mode works great on my tablet and works way better.
Quite frankly, I can’t see ST bringing this to general release because there’s a lot still to be done to align with how the new mobile app works/looks, as well as if this is the right use of developer resources to work on vs. fixing the new app and making that better first.