Be aware that those extracts are squares around the cities - they're not exact city border outlines.
This means that you can't just use the data without concern for whether the things you're gathering are actually within the city or not.
Some cities make shapefiles available for urban growth boundaries and city limits. It's worth enquiring. But be aware that some cities city limits are not simple polygons; depending on local ordinances you can see islands, internal voids and peninsulas attached with zero width stems.
Well remember OSM is an international project, so trying to find one definition of "city" and "border of a city" is hard. Some places (like USA) seem to use "city" as a hard solid thing, other places (eg UK / ireland) don't.
Except that the place tag isn't always based on population size. In some places, (e.g. UK), a common cultural definition of a "city" is archaic, and depends if the place has a Cathedral and/or charter from the monarch.
I give you the "city" of St. David's: http://www.openstreetmap.org/browse/node/3216768 It's in Wales, UK, and is tagged as "place=city". It's a small town of ~2,000 people. It has a charter from the monarch, and a cathedral. The "note" tag in OSM says: "the "city" is on paper, what's on the ground is a small town".
Greenhorn, Oregon is designated as a city but has a 2010 census population of zero. Though now unicorporated, Tenny, Minnesota, was a city and had a peak population of 180 people in 1910 but now is down to 5 (2010 census).
I've been playing with place names & http://overpass-turbo.eu/ ... I seem to be getting the best results with "city|hamlet|metropolis|town|village", although some states like Alaska & Hawaii still don't return much with this scope.
In Alaska, a lot of places are labeled as County. In Hawaii, a lot of places are labeled as Other.
I haven't found a very simple way to get everything within exact city borders, yet. This is the process I've been going with so far: https://github.com/JamesChevalier/cities#cities