Of the people in the examples in the link, only person #4 is clearly cyclically unemployed. People #2 and #6 might be cyclically unemployed, though person #2 would not count in official unemployment statistics.
Cyclical unemployment is a kind of unemployment that occurs when a person loses their job because the economy has gone bad. Consumers still want the product the person makes, and their job has not been replaced with machinery. The problem is that people lack money during a recession and can no longer buy the product.
Person #4 is clearly in this situation since the economic downturn is the cause of her lost job. With #2 and #6, we are not told why they are out of work. If they are out of work because of the economic downturn, they are cyclically unemployed. But #2 is not going to count in official statistics because he has quit looking for work and is therefore a discouraged worker not in the labor force.