correct. very few cars also have the feature that turns headlights/taillights on when you turn the wipers on.

no vehicle has an auto mode that turns on just your headlights. full stop. it doesnt exist. that would be insanely dangerous. if thats what is happening in your car your tail lights are burnt out.

you also can not turn off your DRLs. there is no suuch thing as "auto drl". you either have them, or you dont.

on some toyotas the drls are the parking lights. thats how my tundra is. instead of a dimmer than normal headlight, its a brighter than normal parking light.