U.S. jobless claims surge to 26-year high (highest since October 1982)

I won't refute you on that (someone once posted a article here on HN that discussed this) however the linked article for this thread is specifically about people filing for unemployment for the first time. Thus your point that "They stop counting a person as unemployed after they stop receiving the unemployment checks, no matter if they found a job or ran out of time" doesn't taint the stat of a 26-year high for filings for new claims.

Well, yes and no. This article isn't about unemployment, it's about jobless claims. While jobless claims are tallied the way you describe, unemployment numbers are derived from a monthly poll:

http://www.bls.gov/cps/uiclaims.htm

Those numbers also probably understate the problem, but not for the reason you mentioned; the official unemployment numbers do not include people that are underemployed or those that have stopped actively looking for a job.