So first of all, any significant drop in employment is overwhelmingly likely due to the GFC.
Second, I must take issue with the idea that they (the administration) "redefined" the metric. As I said above, there is a genuine (and basically settled, I believe) question in economics about what the right way is to summarize the state of the economy's labour force in a number. So any concern with it is going to have to be a technical concern: does a measure summarize the state better or worse than another measure. In fact, the BLS consists of professionals with a specific job to do, so I would be really skeptical of a claim that the administration redefined a metric that was inconvenient to it. It is rather more likely that it was already settled on economics grounds previously.
It would be really surprising if the BLS, an independent statistical agency, did something like what the argument accuses it of doing, so there needs to be some more solid evidence that it actually happened.
Specifically with respect to people who gave up looking for work being included as unemployed, I'm pretty sure that's been like that for ages now.
Third, and my main problem with that argument, is that it's an argument ad hominem. Instead of settling the choice of unemployment measure on merit (economics is a relatively old discipline, there is more than enough to talk about to decide what the merits are), the argument instead tries to argue based on imputed motivation of the current administration. That's a very dodgy road to go down, because settling it on merit is always a better thing to do.
> these cannot both be the case
It's not my main field of expertise, but this, I believe, is just plain wrong. The structure of the economy can definitely change with time. Two prominent examples to my mind are: (1) women entering the labour force, leaving the unemployment rate unchanged, but drastically increasing the number of working adults, (2) changes to disability and health insurance rules that sometimes mean that people who are seriously ill drop in/out of the labour force with good reason (they might be very sick yet still need to work to be able to afford things). So I think it definitely can be the case that different measures will go different ways, depending on the underlying changes in the economy.
> Re people dismissing clear facts, again, this is just human nature.
Meh, I don't care, I just want to get it right.
Edit: On further thought, I think the argument you mentioned is pretty damn close to dismissing statistics. Your original argument, above, was totally different, and much better.