Actually if you are seeking medical treatment for Steatodism it does help to bring the, doubtless now squished, spider with you to prove that you're not suffering from something else that can cause a severe allergic reaction.
The venom of a Steatoda is a neurotoxin not a necrotic agent. Seeing a spider that was in a different part of the house at the time of the bite/sting is not sufficient to identify the symptoms as being from a Steatoda bite.
As has been said, these guys usually bite once as a defensive measure and certainly S.Nobilis doesn't hang around in tight groups because they tend to kill each other if they get too close to each other's territory.
The most common house centipede, S.coleoptrata, stings with it's legs so you're likely to get more than one sting from one of these guys. Their venom also causes more localised pain and swelling without the neurotoxic effects of the Steatodae.
So, yes, unless you see the spider that bit you it's usually guesswork.