I have four little girls. They are all toddlers and I’ve learned toddlers will do anything to be like their mom and dad. I believe it was just your son wanting to be like you. Stuttering is not something you can completely avoid. Sooner or later your son will find out you have a stutter. Why not open up and talk to him about what it is (when he’s able to understand) instead of try to hide it? I’m sorry your feelings got hurt by this. Try to remember that he is just a kid who doesn’t not understand.