Hi Number42, yes, that is still home education. I agree that Edyourself is a fantastic resource for this sort of information.
After the end of compulsory school age parents no longer have a legal duty to educate their children, and the legal responsibility falls on the young person to remain in education or training until their 18th birthday. Home education is a permissible way for them to meet this duty.
Local Authorities don't take any interest in the quality of elective home education for young people who are over CSA, so you won't be hearing from them with a request for details about the home ed. They may ring or send a form to ask your young person whether/where they are being educated. Currently this is just for statistical purposes: LAs are supposed to offer help to young people who aren't able to access suitable education, and the government will be asking for their figures. So far there are no criminal penalties for failing to continue in education anyway, so your child won't be asked to prove anything.
In practical terms, the sticking point is that you can't claim the usual benefits (Child Benefit and Tax Credits) for a young person continuing in full-time education, unless they had been HE at some point before their 16th birthday. (I think the government fears that everyone will jump on the home ed bandwagon and claim to be home educating in order to claim benefits, when they aren't really doing anything, so they use "track record of home ed pre-16" as a test. It seems unfair to me, but that's the rule.)
If finance is an issue for your family, I wonder whether distance learning via a provider such as InterHigh might qualify you for benefits, just as attendance at a bricks-and-mortar college would? I don't know whether it does, so you'd have to check. But you do have to pay for the education in this case, so that might not leave you any better off...