Counting reinfections

People testing positive often count once, so what about reinfections?

The ITV political editor Robert Peston claimed on Twitter:

In case you didn’t know — and I didn’t till a senior government official told me — the daily tally of infections seriously understates the actual number of infections, because if you are sick with [Covid-19] today but had it any time in the past (even last spring) your new bout is not included in the daily dashboard figures. We know people are being reinfected.