The 4 types

"Ladybird is what they're commonly called in the UK
Ladybug is what they're commonly called in the US"

Right.... it's actually not a bug at all it's a "Ladybird Beetle" (family: Coccinellidae).
They're both insects but true bugs are members of the order Hemiptera, while beetles belong to the order Coleoptera.

I think ladybug makes more sense, but as a Brit I have to stick with ladybird.