Is tax free cash from your pension classed as income

I have savings and understand that I have a personal savings allowance of £1,000 which allows me to earn up to £1,000 interest per year tax free. However I believe that if your income is under £12,570 that you have an additional savings allowance of £5,000 (see https://www.gov.uk/apply-tax-free-interest-on-savings), and this extra allowance drops by £1 for every £1 you earn over £12,570 Is Tax Free Cash taken from a pension considered income? Specifically, if I draw down £12,570 from my pension as income and I also take £2,000 as Tax Free Cash from my pension is my income £12,570 (so my savings allowance £6,000) or is my income £14,570 (so my savings allowance is £4,000)

Hi,
The tax free amount does not get included.
Thank you.