Spiders are bugs! But not insects. (You are right, critter doesn’t work here)
Oh you just had to start discussing arthropod terminology, didn't you?
In entomologist-speak, "bug" refers to true bugs (Hemiptera, e.g. stink bugs, cicadas), which is an order of insects, so spiders aren't really bugs, either. But English seems to lack a common term for all these creatures. Maybe "creepy-crawly"? Not a
word, as such, but a term that covers all those things with more legs than many people can cope with. One could use "arthropod", but that's more technical.
Or you can ignore the pedantic entomologist and continue calling them all bugs.