I used to live in one of those upscale areas you mentioned. The Watchungs are a pretty area.
For what it's worth, Union County is counted as being in the NYC Metropolitan Area by whoever looks after such things, as are Middlesex, Monmouth, Ocean, Somerset, Morris, Sussex, and Hunterdon Counties from your list. Including Hunterdon in the NYC Metropolitan Area might be a stretch -- it has some rural areas as I remember.
The census categorizes only urban and rural. Their criteria is solely population density (urban = more than 1000 persons per square mile, rural is pretty much everything else).
Unfortunately the census definition doesn't really fit well with the standard traditional definition of urban which is more of what most people think of when they hear the term and which mentions nothing about population.
The census definition allows a general area (for example a township or county) to be categorized as urban even if it mostly farm fields or woodlands. As long as there is an average population density of over 1000 people per square mile. Conversely a large industrial area with little real population can be designated as rural.
The census also makes no consideration for suburban, areas that have higher then 1000 person per square mile populations but that are not really anything akin to a city. For example, a square mile contains 640 acres of land. If you divided that into 2.5 acre parcels with a family of four living on each parcel, you would have a population density of greater than 1000 people per square mile. Now I don't think you would find too many people that think that one home per 2.5 square acres equals an "urban" landscape but the census thinks it is.