Tow package confusion

Hey fellas. I know there are a million threads about this and I've read most of them the past few weeks but can't seem to find an answer.

I bought a 2012 tundra grade, 4x4, 5.7, my first truck. Searching by Vin it does say that it has the tow package. That's where the info gets confusing.

I've read that that all 5.7s have the tow package, that they all have 4.30s, that there's a tow package and max tow package, and I'm lost. What threw me off is that I have a class 4 hitch but don't seem to have a transmission cooler.

My door jamb sticker says B02A/AB60F, not sure if that helps seeing that there are inconsistencies and confusion about that as well. I'm gonna jack up the rear to test out the gear ratio anyways but wanted to see if I could get things cleared up.

I appreciate any help!
Thanks

 

If you don't have the Tow/Haul button and you don't have the transmission cooler, then you have the 4.10 gears and no towing package. You would also be missing the trans temp gauge on your cluster.