Actually, they do, and it is the one you hear most often in the media. I'm not sure where this fiction came from that people off of unemployment aren't counted among the unemployed, but the only three criteria for being counted as unemployed are:
- That you do not have a job,
- That you have actively looked for work in the past four weeks, and
- That you are currently available for work.
I've noticed a disturbing trend lately, mostly from right-wing nutcases, to try to redefine "unemployment" to be something that it's not, in some way that is different from how it's been calculated for decades, to include people like retired people not seeking a job, students, new mothers who have voluntarily left the workforce, people who haven't sought a job in more than a month, etc.
Unfortunately for them (and you), unemployment has a specific economic definition and doesn't change based on what you think "feels right". The current unemployment rate is 5.5%. Arguing that it's something different is like arguing that the mass of an object is higher because your arms are tired and it feels heavier when you try to lift it.