I think your ID is right or at least close. Don't worry about these spiders. They are not of medical significance and the chance of being bitten by a spider is so small that it's almost not worth considering as the source of your child's skin issue.

You didn't mention your location. I can give you a list of spiders to watch for if you tell me where you are. But in general you need to watch for the brown recluse and the widow spiders. Familiarize yourself with those spiders and then you don't need to worry about the other ones.

If you do find spiders inside just scoop them up and throw them outdoors. Remember that spiders protect us from more harmful insects like mosquitoes!

It looks that way from these images, but the "spot" on the smaller one is the actual spinneret, whereas the spot on the larger one is a distinct spot above and separate from the spinneret.