Welcome to YT. Sorry to hear that your little one is sick. Like Maximo said, don't believe everything that you read online about well.....just about everything including dog food.
Glad to hear he if no longer throwing up and back to drinking water. If you are going to home cook in the future there is a whole lot more to doing so than just tossing together people food. To do it properly you need to feed a properly balance meal with added purchased supplements. There are several Vet Nutritionists that you can purchase or even use for free their properly formulated diets to home cook for your dog. But for now, you need to concentrate on making sure that his is back to feeling well and eating what he is used to being fed. He is still very young and has a delicate tummy that is just getting used to even eating puppy food.