This breaks down in Australia where the spiders are as big as your head or in Puerto Rico where the foot long centipedes eat bats. Those aren't bugs, those are nightmare fuel.
But the OP is right, of course. Spiders are not insects. Although I suspect that most people I'd talk to in the UK would call them such. I'll have to hand in my Pedant's Society card (which is actually made out of plastic)...
The American use of "critter" also bemuses
"terrestrial arthropods, such as centipedes, millipedes, scorpions, and spiders, are sometimes confused with insects"[1]
[1] https://en.wikipedia.org/wiki/Insect#Phylogeny_and_evolution
In real life many people call spiders insects, either from ignorance or from not caring about the distinction.
To refer to both insects and spiders & similar in one term you could use the (rather less formal) terms "bugs" or "creepy crawlies". I can 't think of anything more formal off the top of my head.