Should I reserve my "emergency savings" solely for times of unexpected unemployment, or keep these savings flexible enough to cover other unforeseen critical needs?
Emergency funds are good to keep yourself out of debt, for whatever reason.
Job loss is a big place where an emergency fund can help you out. It buys you time to find another job before hauling out the credit cards for your groceries, falling behind on your mortgage and car payments, etc.
But it can just as easily be used for major car repairs, serious medical issues, home repairs, etc. ... anything that needs to be done quickly, and isn't a discretionary item.
The bigger your cash reserves, the better, especially now that the economy is bad.
I think it is stated perfectly in the question, "unforeseen critical needs." You know you will need to buy new tires for your car, they are critical but not unforeseen. However, if a tree falls on your car and you need to pay the insurance deductible for the repairs it would be unforeseen. You should budget for the expenses you can plan for in advance like car maintenance and repairs. An emergency fund is for items that are out of the ordinary.