Yes, you pay less tax overall, but the reason for that is that the money is treated as having been given to the charity pre-tax. It does not make your net income go up.

In the illustration in your link, Sue gives 1k to charity, and as a result, and pays 350 less tax. This means that, as a result of this donation, her net income has fallen by 650.

I did not provide the specific names of all the rules, but I thought that would be obvious from the numbers, context, and some of the terms

I had not considered the difference between payroll giving (my first example,give 1k before tax, charity gets 1k, your taxable income is 1k less), and claiming back (you give 1k after tax, charity gets 1.25k, you reclaim some of the tax you paid), but payroll giving is more efficient for the donor, as they pay zero tax on the donation, rather than basic rate, as in the reclaim method.

There is no UK legislation, as far as I know, that reduces your tax bill by more than your donation.