Cars are almost among the lucky exception to the niche rule. However, there was still one group of people in the USA that moved from horses to cars enormously fast — farmers. ... Nevertheless, cars were adopted in cities quite fast as well.
It seems to me that cars became popular in rural areas earlier than in urban areas. I can not find any evidence of that statement. Is that true?
You seem to doubt that cars would be more popular among farmers "in remote areas" because you don't think there would be many people in those areas.
The fact is that at the time that cars were new and farmers were "early adopters," the United States was more rural than urban - with farmers being the most "rural" of all because American farmers tend to live on the farms rather than in towns or villages.
Wikipedia has a table of the urban/rural breakdown of the United State from 1790 up to today.
Around 1900/1910, when cars were first becoming a thing, the United States as a whole was still more than 50% rural. The US became (slightly) more urban than rural in around 1920.
Most folks in the larger cities at the time when cars first became available wouldn't have had a place to keep a car. People living in apartments in cities wouldn't have had their own stables to convert into a car garage, nor would there have been much space to rent to keep a vehicle that would make it convenient enough to be of use. Who wants to take a bus to the other side of town to get in the car and drive half way back to go shopping?
Cities at the time were also expanding public transportation. Streetcars and bus lines covered the cities, making it unnecessary for inhabitants to own their own transportation. (Take a look here, for information about public transportation and read through the following articles to see how cities went from pedestrian to public transportation to automobile dominated.)
Farmers lived largely out of town, and had space on the farm to either erect a shed or an existing shed or stable that could be used to keep a vehicle. There wasn't any public transportation outside the cities. (Yes, there were intercity trains and stage coaches, but you couldn't just jump on a coach to go and visit your relatives in the next town over on a Sunday afternoon. You had to have your own wagon or horses.)
There's not much reason to doubt that American farmers adopted cars more quickly than the folks in the cities.
They had more space to keep an automobile, and more practical use for one than the city dwellers. They were accustomed to having and maintaining their own transportation (wagons and horses) and had need of dependable and fast transportation.
This paper has some relevant quotes about conditions at the time when automobiles were becoming popular.
The need for cars:
During the 1920s, an investigator from the U.S. Department of Agriculture happened upon a farm family that had recently purchased an automobile even though their home lacked indoor plumbing. Asked to explain why, the wife replied: “Why, you can’t go to town in a bathtub!”
The Yearbook of Agriculture for 1928 reported that while eight million farmers and their families lived within five miles of towns of 2,500 or more, fully twenty million still were even more isolated. A government circular entitled The Farm Woman’s Problems revealed that members of the average farm family had to travel three miles to reach the local church, five miles to market, six miles to highschool and the family doctor, and fourteen miles to a hospital. “No burden”, concluded Edward R.Eastman in the late twenties, “has ever set quite as heavily on farming and upon the farm family as has the curse of isolation and loneliness.”