Dehydration kills a great deal faster than starvation. Sunken eyes is a sign of serious dehydration in a child. The child either needs to be fed fluids orally by the spoonful until this improves, which can take two hours of devoted care, or they need an IV. I once treated serious dehydration at home with spoons of fluids to prevent my 4 year old from being subjected to an IV. This is a technique taught in developing countries where IV fluids may not be available.
If a child is vomiting but keeps drinking, they are much less likely to dehydrate, even if it does not stay down. When my youngest son had Winter Vomiting, a deadly flu from many years ago, I told him to drink or I would take him to the ER for an IV. He had had an IV once when he had been sick while relatives watched him.* He didn't want another. So he would drink a whole lot of water and eat a few crackers, go projectile vomit it back up and go back to bed.
* That is not a criticism of my relatives. It is just another example of how much of a difference it makes for a child to have a devoted full time caretaker who knows them well. Children are not good at explaining that they have a serious issue. It is vastly better if there is an adult who is in a position to just recognize that their behavior is seriously off and something is just not right. People who don't take full-time care of a child are just less able to recognize that the problem is serious, even when they are otherwise very competent and devoted caregivers.