As for dictionary definitions, youre right about what the first definition is, but many recognize what I pointed out (insects as animals). Definitions other than the first aren’t wrong, they reflect that people use words in ways other than biologists do.

anything that lives and moves, including people, birds, etc.: Humans, insects, reptiles, birds, and mammals are all animals. Cambridge dictionary

Any living creature other than a human being can be referred to as an animal. Collins

any such living thing other than a human being. Dictionary.com