When displaying cameras and doorbells, the Google Nest app displays "[WHERE] ([LABEL]}" for Google Nest cameras and doorbells released before the fall of 2021, omitting the LABEL if there is none; on the initial screen, the Nest app displays just "[WHERE}" as the label for the camera. However, for the camera on the Google Nest Hub Max, the Nest app displays "[ROOM] ([DEVICE NAME])", and on the initial screen where all the camera feeds are shown it displays just "[ROOM]"..
However, if you change the name of the ROOM (also called PLACEMENT) for the Nest Hub Max--or if you move the Nest Hub Max to a different ROOM--the Google Nest app fails to recognize that change, and persists in displaying the camera with the previous ROOM, even if that ROOM is removed in the Google Home app. Oddly, though, a change to the DEVICE NAME is recognized. The same thing happens on home.nest.com. In other words, you cannot change how the Google Nest app labels the camera on the Google Nest Hub Max.
The change to the ROOM and DEVICE NAME ARE displayed correctly in the Google Home app and on the Google Nest Hub Max and other Nest Hubs. In the Google Home app, when you click on Camera, before a video feed is displayed, it displays "WHERE" for older Google Nest cameras and doorbells, "DEVICE NAME" for the new 2021 Google Nest battery cameras, and "DEVICE NAME" for the camera on the Google Nest Hub Max. On the Google Nest Hub Max itself (and on other Nest Hubs), it also displays "WHERE" for older Google Nest cameras, "DEVICE NAME" for the new 2021 Google Nest battery cameras, and "DEVICE NAME" for the camera on the Google Nest Hub Max (in bold for all), with the "ROOM" in smaller letters underneath; here the modified ROOM is displayed correctly.
I guess we are stuck with this confusion since Google Nest has one foot in each app. But it's a clear bug that the Google Nest app retains the ROOM name for the camera on the Google Nest Hub Max even when it no longer applies and even when it is deleted.
I discovered that if I removed the Nest Hub Max in the Google Home app, and then added it back (including adding it to the desired ROOM), and then set up the Hub Max's camera again, the Google Nest app then recognizes the updated ROOM name. The other oddities in how cameras are identified in the two apps remain.