This is more of an Android Auto question than a Waze question. You may have noticed that other UI buttons are missing, like the App Launcher which also controls split screen, and the Recent Apps bar.
It’s possible Chevy/GM has a different preferred method for launching those features, so has AA disable them for compatibility with their newer interfaces. The controls in our Subaru are like your first screenshot, but we’ve had newer rental cars that look more like your second screenshot, so I understand the disorientation, but don’t have any specific tips for your model.
First thing I would check is whether “Hey Google” voice detection is working or not, or if your wife’s car has a button on the steering wheel to trigger Google Assistant. If you can trigger GA with either of those methods, consider if that is sufficient, and if it doesn’t work, check to be sure Android Auto has permission to use your microphone.
If that’s not sufficient and you really want that button back, I would dig into the Equinox infotainment center settings, and the Android Auto settings on both the display and your phone. This Google Help page might be a good place to start looking, but you can also check user forums for your year of Equinox for tips, or your user manuals.
Good luck!