the BBC isn't a public body in the sense that is, say, the British Army. The Army is funded by a general, compulsory taxes on income and other trade. The BBC is funded by a licence which you only need to pay if you choose to watch (possibly time-shifted) live broadcast television
A tax doesn't have to be universal, unless you're also going to argue that the tax on cigarettes and alcohol aren't really taxes because only smokers and drinkers pay them.
The licence fee is a compulsory tax on anyone who watches broadcast TV, whether or not they consume or even care about BBC services. Now I'm not saying that I don't enjoy BBC output, or even that I necessarily resent paying the licence fee, but please don't try to use weasel words and pretend it's something it isn't. It might be a special purpose tax and the money it generates might be ring fenced, but it's a tax and the BBC is a public body.