EV = “Use battery first”. HEV = “Maintain the current charge.” I think the goal is to use 100% of the battery, as long as you can charge it up again. If you are driving further than the battery range before charging again, it’s most efficient to use the battery/EV mode at slower speeds/stop & go driving. So on a longer trip, PHEV drivers will try to use up the battery on the slower sections and switch to HEV on the highway/faster sections of the drive. But really only need to do if you want to save some of the battery for the end of the trip, when there might be slower driving. Otherwise, just leave it in EV to make sure the battery gets used up. However, it helps to have extra charge in the battery for any hill climbing on the highway. Otherwise, you are stuck with the noisy ICE and won't have as much power climbing up mountain passes...
Auto mode is a bit of a mystery. It probably does something similar - Tries to use EV mode at slower speeds and then HEV at faster speeds. But it doesn’t know how far your total drive is, so may not use up the entire battery. Unless maybe it takes input from the onboard Nav? That way it knows how long the total trip is and where the slow/fast sections are. But that’s still unverified. In any case, Auto is best for long trips when you don’t want to bother with switching EV/HEV modes. “Set it and forget it!”
The PHEV will start the ICE for all sorts of other reasons, besides an empty battery. Mainly for heat, but also for extra power and to keep the engine up to temp and ready to cut in at any moment. If not for power, it’s usually just idling and not using much gas (EV light will be on at the same time ICE is running). It helps to watch the hybrid diagram to figure out what it is doing.
Secret tip: Switching to Sport mode on the highway will charge the battery! Not a lot, but if you are on a long trip and have used up the battery, Sport mode will charge it up for some slower driving after the off ramp. We drove cross country and would use Sport mode 15 min before an exit. Gave us 5 mi or so of battery charge.