The Caffeine contained in coffee can sometimes have a laxative effect on some people so they would naturally have diarrhea or have to have a bowel movement pretty soon after drinking the coffee.
My mom used to get diarrhea whenever drinking too much coffee as well but she slowed down on drinking the coffee and only drank 2 cups of coffee per day and she stopped having diarrhea.
Still some people may experience diarrhea with only one or two cups of coffee and that's normal as well but not everyone experiences the diarrhea when drinking coffee.
I never get diarrhea myself after drinking coffee and I drink around 6 to 8 cups of coffee per day but the coffee does make me pee a lot.