'Bug' in that context is mostly American dialect. I'm British, and 'bug' in that context is infantile; by adults, it's either a verb or a small device planted by MI5.
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)...
And also another source of British vs. American dialect difference/confusion. In my experience, they're known in the U.K. as "woodlice", but in the US it's entirely reasonable for grown adults to refer to them as "roly-polys"
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)...