No, my pension is government from NYC/NYS...
Well, pardon the semantics, but I consider that still private, as opposed to Social Security. Private in the sense that Democrats were buying votes by showering teacher unions, etc. with lavish pension promises. As opposed to the federal Social Security which is totally ignorant to your political affiliation or anything else but your earnings record.
So, if you insist on technicality, you are right, it is a government pension. Buth chances that a city like NY or Chicago could go bankrupt are indeed much higher than of the federal government.