There are various ways to handle the point system. Either way is fine. The problem with the first method is that it eliminates children from the group activity. The problem with second method is some children will make mistakes on purpose because they want to be the leader. This is a fun game.
Children enjoy playing on their own.
But when an adult is the leader it presents an opportunity to teach children about animals they may not know. The adult can yell out uncommon animals and insects such as an Emu, Orangatan, Cicada, Slug, etc. which are sure to elicit mistakes from the group. One child is chosen from the group to be the Guesser. The rest of the children make a circle around the Guesser. The Guesser is blindfolded and spinned around clockwise by the adult a few times.