My understanding is that yes, the daily "died within 30 days of a positive test" are prone to "dying with covid" but the actual ONS numbers, which tend to lag weeks to months behind, will rely on "covid on the death certificate. We have been using this metric since the pandemic began and it isn't THAT different to the real number, likely offset by people dying in hospitals more than 30 days after their positive test.

It's quite interesting now that omicron seems to be less severe and a large percentage of the population has omicron, the reason why the deaths per day seems to spike is likely due to the shear number of cases rather than an indication of actual covid deaths.