Since the common cold is caused by coronaviruses a decent chunk of the time, I'm not sure it's even inaccurate for someone with a non-severe case of covid-19 to conclude that they have a cold.
Maybe that's the tricky part that helps it spread. Having a cold isn't enough to stop most people from going out in public. In my experience among people I know in the US and Canada, having a mild cold isn't enough for people to stay home from work.
But while a covid-19 infection might really be just a cold to a large chunk of people who get it, it's unusually deadly to a non-trivial percentage of people as well.
