What type of bicycle is closest to European city bike? I am looking for a bicycle that would be closest to the design that was popular in Russia, something with a high handlebar, relatively large wheels, and relatively short distance from the saddle to the handlebar. Which types of American bicycles should I look for?
"Dutch bike" is a good answer, but it won't necessarily find you all the possibilities; "transpo(rtation) bike" tends toward the Dutch. Besides what you already mentioned in your question, these bikes typically have:


mixte or other step-through geometry
internally-geared hubs for ease of maintenance
simple shifting, typically 3 or 7-8 gears (though with today's internal-hub selection, 11 can be found)
clothing protection, from chain guards and fenders to (less commonly) skirt/coat guards on the rear wheel
racks or rack braze-ons, minimally in the rear but sometimes front as well


US bicycle brands that offer bikes of this sort include Electra (Amsterdam, perhaps the Townie), Civia (Twin City), Linus (Dutchi, perhaps the Mixte), Breezer, and Public (mixte and step-through models). Trek made one called the Belleville that was discontinued (it was a heavy and inflexible beast; I considered buying one) but might still be available used.

In the UK, Pashley makes several models of Dutch bike (e.g. the Princess and Roadster models). A few US distributors will import a Pashley, I believe.