It may be possible that in tornado country, the average Joe and Sally may know the difference between different warning terms — but I think that, in general, knowledge on the topic is slim indeed. How would they learn it in the first place?
See this story on an effort following a tsunami "watch" for the coastal counties in Washington state, Coastal Counties Put Heads Together To Avoid Future Tsunami Alert Glitches. The challenge is that these types of efforts, as well-meaning as they are, will not reach many people.
I think that if you did the "man on the street" interview with people passing by about warning terms, it would be as bad as "Jay Walking" an old segment done on late-night TV.
Are we hopeless or helpless? Is there a difference?