Headlight Issue?

Ok, so it's cold, I live close to my job(s). So sometimes I get out leaving the truck running a n few minutes, then I get in from passenger side shut off and grab my keys from ignition. Usually daylight, so I been missing that my lights are still on. I have my truck lights using the DLR position always.
If I return to driver side and open the door, the lights do go out.
Is this a issue with DRL switch, is this a passenger door issue, with a relay or fuse?
I been trying to make notes the past now 4x's this happened. Has cost me one battery, would like avoid another one. Anyone ever heard of this lights, passenger door thing. Odd but seems to be the common denominator I can think off.
Thanks in advance.

 

If I'm understanding right this is totally normal. My lights don't turn off until I open the driver door either. I'm not sure why, but that's how they are made. I would suggest to just go to the driver side when you shut the truck off.

 

The driver door is the only one that' connected to the relay. Completey normal

 

Normal.

 

Ok. I will relax and stop looking for problem. Thanks guys.