A medical certificate of cause of death doesn’t require watertight 100% conclusive evidence of cause of death. It is what the person certifying thought was the most likely explanation for their death, supported by the evidence they have available. If there is reasonable doubt, it goes to the Coroner for discussion / inquest / post-mortem. That’s not unique to Covid.
The way medical certificate of cause of death is formatted requires 2 parts - part 1 gives direct cause of death (and can have several parts if there is a sequence of events leading to the death). Part 2 are conditions that are thought to have contributed to the death but not directly caused it.
So you might have:
1a) severe acute respiratory failure
1b) Covid-19 disease
- Alzheimer’s dementia
for someone who actually had pneumonitis as their last illness.
But if someone with dementia has a positive test but mild symptoms, goes into hospital and then dies from their dementia a couple of months later
1a) frailty of old age
1b) Alzheimer’s disease
- Covid-19 disease
because you can’t rule out that the covid somehow contributed to their death by making them more frail.
Many people try to be as detailed as possible - so differentiate between “Covid-19 disease (proven)” and “Covid-19 disease (suspected)”.
If there is a positive covid test though it has to be declared, because it is a notifiable disease (like measles, TB etc etc).