Crawdads are still mudbugs though.

But commonly it's a catch all for any creepy crawly, including arachnid. The classification is even called True Bug, not just Bug

The scientific taxonomic system was made, in part, because traditional colloquial terms are a mess. For example, "daddy longlegs" refers to a type of spider in my area, but there are two other animals and three plants that it could refer to depending on where you grew up. Taxonomists saw that there are ten different standards, decided to make a new one to replace them all, and for once, it actually worked out for the most part.

"Bug" is one of those old terms. It might have been mapped post hoc on top of the modern taxonomic system, but it didn't start that way, and isn't always used that way. I wouldn't expect an entomologist to use the term at all in formal contexts.

Merriam-Webster, definition 1:

a: any of an order (Hemiptera and especially its suborder Heteroptera) of insects (such as an assassin bug or chinch bug) that have sucking mouthparts, forewings thickened at the base, and incomplete metamorphosis and are often economic pests

called also true bug

b: any of various small arthropods (such as a beetle or spider) resembling the true bugs

c: any of several insects (such as a head louse) commonly considered obnoxious

I'm sorry but you're simply incorrect. Bug can be a technical term, but that doesn't also preclude it from also being a non-technical term, because words often have more than one meaning. See also: theory.