score:5
Yes, the QR code generated in the NHS app when you have had two doses plus a booster is accepted as a "Super Green Pass".
This is confirmed by the gov.uk page you linked to; I've emphasised the key points:
A Green Pass or a Super Green Pass (see below) is required for the following...
Then below:
Italy will accept the UK's proof of COVID-19 recovery and vaccination record as the equivalent to a Super Green Pass as long as it is in the form of a verifiable QR code.
And:
Once in Italy, a primary vaccination certificate (demonstrating both doses of a two-dose vaccine or one dose of a one-dose vaccine) serving as a Super Green Pass has a validity of 180 days from the date of the last vaccine or is valid indefinitely for those who have had a full course plus a booster dose.
It's a good idea to save a PDF copy, and maybe print some out, so that you're not reliant on your phone and data connection.