To my knowledge, this is not true in the US at least. Folks would just donate whatever they owe in taxes to their church or favorite charity if it directly reduced owed taxes.
If I donate $1k, it reduces my taxable income by $1k, saving me ~$300ish.
No, charitable donations are deductions, which mean they are deducted from income. What is deducted directly from tax liability is credits, but charitable donations don't give 1:1 credits.
Oh, really? Does that mean that, if you owe $1000 in tax, you can give it to a charity of your choice, rather than the government? That sounds like a pretty big advantage.
In canada, anyway, you have a limit as to how much you can donate and you receive tax credits in return. It's not 1:1 though, so it's not as if I can just donate my taxes to a charity rather than the government.