As Paladin is wrapping the rein around the stake it is turning meaning it is so loose that Paladin could have easily lifted it out.
When Paladin removed the snake from the tree his finger and thumb are directly behind the head. This prevents the snake from twisting to bite the hand holding it. But as Paladin walks along with the snake held above his head, his hand is now 3"-4" from the head, more than enough for the snake to twist and bite him.