Sara, nice job summarizing the situation on UM which most people do not understand.
There is another part to this, if you choose to fight, some insurance companies and start have what they call a full tort option, if you have full tort, then you over insurance rate is higher since it allow you the option to sue. If you do not have full tort option then you can not sue, and you agree to accept what your insurance company choose to pay out.
As I tell people when they are trying to decide which way they should go with all of this. I tell them how luck do they feel, to quote Dirty Harry. If you choose minimums and such and you get in a serious accident and the other person can barely afford the car let alone the insurance you are probably going to be in serious hurt.
What people do not realize is collision and comp are the only part of the insurance that covers your car, all the other insurance cover the other person whether it is your fault or theirs. Liability is if it is your fault and UM/UIM is if it is their fault. Also some companies also include Medical which is to cover your medical deductible of your personal health insurance, This cover any expense that your personal health insurance does not cover.
As one person pointed out, none of this covers lost wages, if you in a serious accident and you work for a company, usually you can file for disability insurance to cover you lost wages if you are that hurt that you can not work.
So again, how luck do you feel not to have coverage, considering the worse drivers are the same people who do not have lots of coverage nor do they care if they loose what little they have.