As I've asked the same question without recognizing this one, I put my thoughts here, too (and add some more things).
Both diet and nutrition are synonyms, I would prefer nutrition as the main tag as it sounds more professional and doesn't carry the connotation of weight-loss.
feeding should be perceived as the actual process of feeding (and methods used during the process). Which would have to be noted in the tags wiki excerpt.
The tag excerpt of feeding already describes the desired use of the tag, I currently disagree to make eating a synonym. Eating actually refers to the process of the animal eating stuff and feeding refers to the owner offering the pet some food, if the tag wiki excerpt of eating is edited it could make the distinction clear.
diet and food should be made synonyms of nutrition, the tag wiki excerpt of nutrition currently states: "Refers to the question regarding nutritional aspects of diet."; diet reads: "Refers to the question concerning about food diet of their pets."; and foods excerpt is emtpy.