What determines if a lizard will increase my maximum stamina?
Solution 1:
Yes, it has to be a lizard with a shining white tail.
The normal (black) lizards will have no effect on your stamina.
In a similar manner, eating fruit from trees will increase health.
Hunting lizards and fruit is a understandably a common pursuit in the game, see here for help if you're looking for it.
The bow is recommended for both pastimes.