Vegetarian: no animals killed for food, meaning, you can eat products “made” by animals, like eggs or dairy but no flesh of animals.
Paleo: no foods that according to evolutionary theory our paleolithic era ancestors didn’t eat, like dairy, grains or beans, processed foods (sugar etc.), diet consisting of whole foods and what you can make from them, animal products are encouraged and consumed.
Ketogenic: animal products, dairy, are totally fine. The main thing on a ketogenic diet is to eat foods maintaining the principle of high fat/low carb/moderate protein diet, roughly 75–90% of calories is fat/15–20% protein/5–10% carbs. Eliminating any foods that raise your blood sugar and kick you out of ketosis, a state when your body burns fats for fuel instead of glucose producing ketone bodies (water soluble fat molecules), foods like grains, sugars of any kind, fruits, starchy vegetables and beans, but even too much protein – the ratio of macro-nutrients is the KEY and also not consuming too many carbohydrates overall, for many people the amount of carbs is 50g a day or so.