> There is no redistribution. Everyone is entitled to the same benefit, just like they are entitled to fire protection.

This type of thing (even when it involves fire protection) is redistribution. One set of people are taxed to pay for things, another set of people get the benefit -- even if it is the same set of people the distribution of the tax and the benefit aren't the same, and the difference is redistribution.

Note that I am not saying this is undesirable, merely that it clearly is redistribution.

>One set of people are taxed to pay for things, another set of people get the benefit

Ah, no. ALL people pay for things. How that payment occurs is a product of a progressive tax system in most cases, but there is no requirement that such a system exists to run service.

You wouldn't call a private security service employed by a gated community "wealth redistribution", would you? Although if they had an income driven progressive fee structure it would be.