Both should be accepted here; they weren't before but I've added that now. The general idea is that det is often it but can also be that (in most cases, both should be accepted answers in English).
that can be det or det där (depending on the Swedish sentence: in many cases only one of them works in Swedish)
det där is always that
this is det här or detta and vice versa.
"This is definitely not a spider" is too far from "in any case, it's not a spider" to be accepted? Is "in any case" to be understood as "anyways"? I'm not a native speaker, but "in any case" doesn't sound idiomatic to me unless it's a somewhat defiant answer to a spider- related topic.