When I ask whether someone's living situation is rural or city, what's the category I'm asking about? Living situation includes many other things, but there's gotta be a word summarizing those two things, right?
Or Population density. (Advantage: quantifiable)
The above are more often (perhaps exclusively) applied to a place rather than a person. That is, they are geographic terms, I understand you want a demographic term.
Great question. And timely as the world recently (2007) crossed into 50% urban. Some more ideas, with tongue half in cheek, for a single-word term:
Okay I'll stop. The U.N. seems to call this demographic dimension "urban/rural".
I noticed during a elementary-school research project that kid happiness seemed to increase the further I drove from D.C. (the more rural direction of your spectrum). Also increased were noise and freedom and agedness of clothing and buildings.
I don't know of a special word to describe this relation. From a geographic data perspective, the more accurate phrase would be "urban" versus "rural". The US Census Bureau has a very precise definition here where they call it simply the "Urban and Rural Classification".
If you are meaning urban, rural and so on, the only thing I can really think of to describe them is Areas. This could lead to some confusion though, as if someone asked me what Area I worked in I would say Trafford park or Computing, not an industrial area.
I think this is one of those times when the oddness and ambiguity of the English language shines through!
If you are talking about a heading to put on a column of a report, where you are supplying the answer, then I agree with BobStein that "Urbanization" would be a good choice.
But if you are looking for a label to put on a box on a form that would lead the person filling in the form to answer "rural", "city", or "urban", I don't think there's any choice of words that would get the desired response other than listing the choices. If I was filling out a form and came to a box labeled "Urbanization", I'd have no idea what to put there. I think your best bet would be to give the user a multiple choice question, where the choices are "urban", "rural", "suburban", whatever. If it's a form on a computer, give them check boxes or a dropdown; if it's a paper form, give them a set of check boxes or have them circle one or whatever.
Even if you do find some term officially sanctioned by the census bureau or whomever for this idea, it's unlikely that most of the people filling out the form would know what it meant.
If you want a category where the members (not subcategory) are 'urban' and 'rural' (and I guess additional elements like 'suburban', 'town', 'village') it would be:
density of settlement
or more generally
type of settlement
(the latter allows variation: a village may be more dense than a suburb, but also more rural).
As an aside, it looks like you desire a single word. There's no guarantee of a single word...ever. The description may be the best there is. But there usually is a minimum succinct one.
Also, there may be an official term which is in some sense technical (meaning stipulated by authority and whose natural meaning may be imperfect for the concept or even irrelevant: e.g. 'metropolitan statistical area').
Or one can also just name the category with a (hopefully) short list of members (e.g 'urban/rural').
I'd say "urbanicity" (although I note that someone on the internet is claiming to have copyrighted a process for determining this, so not sure what implications this may have in using it in a survey) or "rurality", depending on what question you are asking. I'd consider the two terms to be inverse to each other (i.e. high urbanicity = closer to urban area, high rurality = further from urban area).
I'd tend to use urbanicity more than rurality, as usually when measuring things, it's in relation to urban areas (as rural areas are more geographically diverse). Of course, if you were measuring proximity to some rural aspect, then rurality would be more appropriate.
Not sure if these are official terms, but I'm sure I've read them somewhere in geographical literature.
It would be nice to have one common term though.