"As I understood 'I stopped by Tom' is not correct, is it?"
Not correct for what you want to say. It would only be correct in a specific situation. For example, if you were in a room and Tom was standing or sitting still, and you were walking past and stopped. You could then say that you stopped by Tom. In that case, "by Tom" is simply a location like "by the window".
You want to say "I stopped by Tom's [place]". Notice the possessive "s" ending. You are implying a place not a person, even though you are using his name alone.
The constructions using "call at" or "call in at" imply a more social visit, whereas "stopped by" is more general and may or may not involve socializing. For that reason, we probably wouldn't say "call in at" a place if we are only going there to conduct business.
Also, I could be wrong, but I think "call in at" or "call at" is a bit stiff and formal sounding even in Britain. "Stopped by" or "dropped by" sound more friendly, at least to my ears.