This is correct about rented properties, but not about empty ones.

Empty properties can attract a discounted rate of council tax, but are only exempt under certain circumstances (resident in prison/hospital, house unlivable).

Long-term empty properties can even attract a higher rate of council tax.

That last part is simply incorrect; Councils have a time limit on whether empty properties are exempt from council tax. Also, there's an extra 50% penalty if it's been empty/unfurnished for 2 years or more: https://www.gov.uk/council-tax/second-homes-and-empty-proper...