Source: https://sundhedsdatastyrelsen.dk/-/media/sds/filer/rammer-og...
This is one of many reasons that it's difficult to compare death figures from different countries.
As for the cause of death, FullFact has this to say:
When completing a death certificate, the responsible medical professional is asked to determine which conditions may have contributed to death. They are also asked to determine “the disease or injury that initiated the train of events directly leading to death” and this is called the “underlying cause”.[0]
So deaths that were caused by COVID-19 will be reported as such, and that's different from the number of people that died with COVID-19.
Note also that it's a criminal offence to knowingly report an incorrect cause of death on a death certificate.
[0] https://fullfact.org/health/causes-of-death-october-2020/
1) Died within 28 days of a +ve test
2) Died within 60 days of a +ve test (in England, but not elsewhere)
3) Death certificate mentions covid
4) Excess deaths above that expected
Then there's the "date reported" and "date of death". The date reported is the day the notice reaches the death registrar. That can be within an hour or so of actual death, or can take months. Most are done within about 5 days.
For November, in the UK, there were an average
418 deaths occurred per day in the 28 day after a +ve test and 396 deaths reported
410 deaths per day on the death certificate
In England alone
28 days: 354 (occurred), 335 (reported)
60 days: 380 (occurred), 358 (reported)
Excess deaths: +251 (5 year average), +310 (vs minimum), +191 (vs maximum)
If the majority of those deaths were people being hit by a bus, you'd expect the 60 day number to be about double the 28 day number
Only if the testing was unrelated to the death or hospitalisation. If people who get hit by a bus are routinely tested in the ambulance or in hospital, the outcome will be more likely as reported.
In November in England, 354 a day died with a positive test within 28 days. 1635 a day died anyway, meaning 1281 died without a positive test.
If 1% of the population were positive, that would mean 16 people would a day "die with covid".
To get 354 "with covid" deaths would require 20% of the country to test positive.
Mass lateral flow testing in Liverpool last month (Liverpool being hit far worse than most of the country) showed 0.6% of the population with covid. Reports were they were pretty bad flows, missing about half the actual cases, so lets say 1% actually had it.
That would mean that of 1635 people a day dying, 16 would "die with covid".
Many other stories like that
You could, for example, read the story you linked, which states: "Despite health officials knowing the man died in a motorcycle crash, it is unclear whether or not his death was removed from the overall count in the state."
So it's a nothing story, with no statement of fact included at all. Just "unclear whether or not" against repeated statements by state officials that "not."
We can rule out the "they're reporting everything as COVID and that's making the numbers high" hypothesis by looking at overall excess deaths - how many more deaths is the nation experiencing than is typical. A misdiagnosis won't affect excess deaths, as it counts all deaths, regardless of cause. The CDC provides a page for this:
https://www.cdc.gov/nchs/nvss/vsrr/covid19/excess_deaths.htm
As is readily visible in the chart, we've been substantially over the norm since late March.
(Switch the chart to New York City for an even more obvious example of how a spike in excess deaths shows up. Screenshots: https://imgur.com/a/zSkNbxT)