"Background checks stop criminals from getting guns" is not inaccurate, it's an absolute fact.
It's not my fault if people read that statement as "Background checks stop all criminals from getting guns." I shouldn't have to clarify myself just because certain people (not you) are getting all hot and bothered by things that I'm not saying.
Your experience with online gun sales is interesting. And the figure about 40% of purchases occurring without a background check did indeed only have 251 answers out of the 2,500 households asked, so I'll agree it's a pretty terrible number. Not completely useless, but also not completely useful.
You didn't answer the one question I was interested in most. Do you support background checks between unlicensed private buyers and sellers?