Nowhere is it noted that he was believed to be under the influence. He was not charged with a DUI, nor is there any suggestion that he even underwent a field sobriety test. Not sure how this is done in Alabama, but they perform a similar test to that which they perform for drinking if they suspect other drug use up here. In a case like this, if you fail the field sobriety test and they smell weed and find it in your car, you get charged with DUI. With a really good lawyer you can beat the charge, but it is hard.
Cotton was driving, the officer smelled weed and found weed when the car was searched. That is all that we know for sure. We do not know that he was driving under the influence or weaving in the road or anything unsafe.