If you used to make 80k in 2008 and now make less than 10 dollars an hour working two jobs, you’re not counted as unemployed but you are very like underemployed.
If you lost your job in 2008, lost your house in 2009, and became homeless in 2010 and basically stopped looking for work, you’re not counted as unemployed because you only count people who are looking for employment, which excludes basically anyone who just gave up.
Usually unemployment rates are seasonally adjusted so you don’t think that suddenly everyone is employed in November and December just because they have temporary employment for the holidays. Same with agriculture.
There’s some debate about people in part-time work because it’s not easy to measure those who choose to be in part-time work because they don’t want to work full-time vs. those who want part-time but can’t find full-time employment as well.
Then you have the expected lay-off group, this group may not be actively looking for employment but they’re in an industry, like logging, that’s expected to have lay-offs for a short period of time and will be recalled. There are rules actually build around these kind of positions when it comes to drawing unemployment too, so they can do that but not be expected to actually leave their industry.
Democrat or Republican, there i a strong argument to be made that the current unemployment rate is under-stated, but it would have been under-stated for a long time, to the point where it would be a poor argument to say xyz group, president, whatever, made it worse.
Unemployment also doesn’t count the prison and jail population. That might seem… silly but with that population growing the way it is and how certain groups circulate in and out of it… that might be worth looking at. I doubt the Republican party would be especially interested in this group though when they reference these other stats.