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?
In the US, such bicycles have the name "Dutch" bike.
I guess, "Popular in Russia" mean "produced in Russia". Russia have never imported European bicycles. This is type of simplified utilitarian bicycle with single speed, kick back brake, fenders and rear rack. The rider position is upright and relaxed.
"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.
Look for a Gazelle or Batavus bike. I know at least Gazelle can be bought in both the UK and the US.
Alternatively, look up what it would cost to ship such a bicycle and buy one in the Netherlands. They are widely available both new and used.
If you are looking for a old one: You will find that old Dutch city-bikes have a better build quality then many other European brands. However, this does not go for more sports bicycles.