In my circle of acquaintances, the distinction lies not in what the money is funding (e.g. health care), but in where it's going:

Health coverage goes to some entity that isn't the government and is usually optional, so it's viewed as a benefit and taken into account when computing total compensation.

Unemployment insurance on the other hand is taken out of every paycheck and handed to the government without an opt-out. Yes, of course you get something for it; the whole point of taxes is that society gets something in return. They're still taxes.