Group State Based on Specific Entity

My understanding of group state calculation is that if any entity in the group is on, the group state will also be on (all the applicable sensors are OR’ed together). However, if you use the “all” option, then the sensors are essentially AND’ed together.
Instead of these two options, is there a way to simply choose a single binary sensor to use for the group state, and ignore the states of the other entities?

Your understanding is correct and no, there is no way to do that with the core group integration.

What problem are you trying to solve / what are you trying to do?

There may be another way.

I have several groups of sensors from an Xbox device (one group per user). Each group contains 4 binary sensors and 4 sensors. Only 1 binary sensor is relevant to the overall group state, so I just wanted to use that one to set the group on/off.
The best alternative I can think of right now is to just remove the other binary sensors from the group, so the only binary sensor left is the one I want to use (at the cost of losing the other sensors’ information in the group details). Or perhaps there is a way to convert some of these binary sensors to normal sensors so they would not be used for the state calculation?

Instead of a group you could create a template binary sensor and add the other information as attributes of the sensor.

Once again, thank you for this information. I will have to start making use of templates in my setup.
What I ended up doing was ditching the groups, and using fold-entity-row to accomplish what I wanted. I just set the one binary sensor I cared about as the head for each entry. This fits what I was looking to do since I really just wanted all the extra information to show up below the main status, hence the groups.
This also solves another issue I was having with groups.