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So they takes "animal" to mean something like "non-human critters". Not everyone uses words intending them to fit the technical meanings e.g biologists give.
Uh, that's not the definition that biologists use. Kingdom animalia includes humans. (along with fish, birds, reptiles, insects, etc.) We're mammals.
Also, a vast majority of people use this definition of the word "animal" when referring to the animals themselves and only tend to use other definitions (which typically ends up referring to non-human mammals or sometimes humans the speaker find distasteful for whatever reason) specifically when contrasting them to so called "civilized" humans.
You can look up the word "animal" in a dictionary and I garuntee you the kingdom animalia style definition will be the first one you see under the noun form of the word with all other definitions (the ones that exclude humans or insects) coming later. Dictionaries typically order their definitions by usage when there are multiple definitions of the same word.
Not the categories biologists use but in the sentence immediately after you defer to the taxonomic classification system, the system biologists use :)
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