I was given a "keep in your wallet" card at my 1st jab, updated at my 2nd jab. It contains the details you reference above... ...not sure why - in a joined up system - anything further is required as proof... and why what was issued at the time would fall short of the required criteria...?
I suspect the card is perhaps a little too easy to create on a PC at home. However, the NHS enables you to get confirmation through the NHS app, or you can get a printed version sent to your home within a couple of days, free of charge by requesting it online: NHS Website
It has a unique barcode and whilst probably not impossible to forge, is probably better than the card you get after the jabs.
Fair enough, but a two-stage system illustrates yet another lack of joined-up thinking & process.
