Joe,
You are at least partially right.
To be counted as “unemployed,” a person must be actively searching for work.
By definition, people drawing Unemployment Benefits must look for work, and, by definition, are unemployed.
Once their UI Benefits are exhausted, however, some people - but not all people - end their job search and drop out of the workforce.
Those people are NOT counted as unemployed.
As a practical question, how does the government count the people who are not receiving unemployment benefits? There are no people getting a check, so how do they keep track of people no longer in the system?