I can't remember what I wrote on that thread, so not sure how this will relate.
Its hard to define a fair share, and I don't think the system exists that would be perfectly fair for everyone. I dont think i can give a definative answer to that question. I think the council tax system as I know it in my area is completely fair. It is based on property size, and I think it's fair to assume that the majority of people live in a property suitable for their needs.
I used to live in a huge 4 bed house which I owned outright thanks to an inheritance, but when I split with my ex I couldn't afford it anymore, even with the council tax single person discount, so I moved to a smaller cheaper property. That is perfectly fair IMO. I bought a very small studio flat with the remainder of the house sale money which I rent out to give my income a little boost so that I can afford to work term time only to be able to be around for my dc.
I am one of those asset rich and cash poor people you were on about. Unsurprisingly, this is one reason I think a tax on second properties would be very unfair.
Income tax is, I feel, a lot less fair. I think the 50% tax rate was a joke, and 45% not much better. I would prefer a higher personal tax allowance and a flat rate for all. Although I know bugger all about economics so I'm not sure how this would work in reality, but it made sense to me when I read someone else's explanation.
So, what would you define as a fair share of tax?