Each employment example you've provided is tracked by the BLS, along with others. Refer to table A-15, or "Alternative measures of labor underutilization"[0] and this[1] release which expands a bit on what they mean. There's also a more detailed FAQ that discusses how unemployment is measured.[2]
There's no single, simple metric that can fully encompass everything when it comes to unemployment. So they rely on six different metrics, which tend to correlate closely. When they don't, it's obvious in the alternative measures . When politicians talk about U-3, they're comparing it to the same measure in previous periods. Politically, the trend is the focus. In no way is this a case of "statistical smoke and mirrors." Brightlines have to be drawn, and U-3 does just that in way that's fairly straightforward and usable by the public at large.
More importantly, trying to focus on U-6 (for example) as your main unemployment rate has significant drawbacks. While these different cases are all related to one another, the economic policy proscriptions for targeting general unemployment aren't necessarily the same as trying to ameliorate involuntary part-time workers (U-6) who want to work full-time and would if the opportunity presented itself.
Finally, BLS also collects wage data by area and occupation[3] that addresses your final concern. It just doesn't get mentioned as much in the press, or by politicians, because wage data isn't easily boiled down into a single number; it's a bit more involved to deal with. It's still used in econometric modeling and to inform fiscal and monetary policy. It's not something that's ignored.
0. https://www.bls.gov/news.release/empsit.t15.htm
1. https://www.bls.gov/lau/stalt.htm
2. https://www.bls.gov/cps/cps_htgm.htm
3. https://www.bls.gov/bls/blswage.htm