Rule for capitalization of geographical areas?

Please tell me if the capitalization of the word "Midwestern" is correct in the following two sentences:

1.) Study participants were all smokers chosen from a Midwestern city in the United States.

2.) Study participants were recruited from six ambulatory care clinics within a large Midwestern US healthcare system.

I had always thought that geographical areas were to be capitalized, but just lost points on a paper for doing so. Is there a rule for this?

Thank you for your time,

Tanders

[Casiopea]

Welcome, Tanders.

the Midwest <place; proper noun>
a midwestern city <direction; adjective>
a Midwestern city <proper name, applies to only one geographical entity>

Here's what the University of Colorado's online Style Guide has to say:

Geographical and Related Terms Geographical terms commonly accepted as proper names are capitalized. Other descriptive or identifying geographical terms that either do not apply to only one geographical entity or are not regarded as proper names for these entities are not capitalized. Cultural or climatic terms derived from geographical proper names are generally lowercased.

the South, southern, southwestern (direction), the Southwest (U.S.),
the West, western Europe, the West Coast,
the Middle East,
the Midwest (U.S.), west, western, westerner