Um, no. There is no “true” unemployment rate, just various indicators of the state of the labor market. Fortunately, these indicators pretty much move in tandem, so we’re not usually confused about whether the market is getting better or worse. But they do measure somewhat different things, and which one you want to look at depends on what questions you’re asking.
After all, what do we mean when we say someone is unemployed? We don’t just mean “not working”, because that applies to retirees, the disabled, playboys on yachts, etc.. We mean someone who wants to work but can’t find that work — a useful notion. But there’s some unavoidable fuzziness about both what it means to want to work and what it means to be unable to find work.
Suppose that I were to retire from the econ biz and take up origami, or something — but could still be tempted to come out and give lectures if offered million-dollar fees. Do I want to work or not? As a practical matter, no, since I don’t get offers in that range. But you could imagine a situation where the numbers were closer, and the question of whether I really want work is genuinely ambiguous.
The usual measure, U3, measures your desire to work by asking whether you have been actively searching in the recent past; it measures your ability to find work by your taking a job, any job. Obviously this can deviate from the Platonic ideal in both directions: there could be people who could find work if they were willing to take the jobs on offer, and there could be people who want to work but aren’t actively searching because they know that at the moment there’s no point — or who are working, but only part-time because that’s all they can find.
U6 casts a wider net; it includes people who are working part-time but say they want full-time work, it includes people who aren’t actively searching but either were working recently or say that they aren’t looking for lack of opportunities. Again, this could clearly deviate from the Platonic ideal, but it’s a reasonable stab at the problem.