To me, the BBC seems a complicated set-up, so I'm not sure if the following is satisfactorily addressing your question...but here goes....

Ask the average Briton, and they would probably say it is state run, via the government. That said, it does appear to be run as a public corporation, perhaps technically owned by the taxpayer. There are shareholders. It is held in trust for the public by the BBC Trust, and funded in the main by the income from licence payers [[about 75% of total income?), and also by the commercial arm, [[BBC Worldwide Limited) which sells the BBC programmes overseas.

If you've not already done so, try googling the BBC website. Also BBC Wikipedia, which I've studied. It might not, of course, be totally accurate, but does give many details about the operations of the BBC, and might be of interest, especially the more technical details.