For the last few weeks, the way it's worked is that I put out food for Lydia and even if Murphy is in the yard, he or she (I'm still not sure) allows Lydia the time to dine leisurely without interruption. Then Lydia, having eaten as much as she wanted, goes back to her house or to washing herself in the yard and Murphy cautiously approaches the bowl. Lydia is fine with that. Murphy is just terrified of The Human Being here, aka me.
Murphy eats a few bites, then scampers off. This is why I've been saying that I think Murphy is being fed by someone else because that cat has never seemed too hungry to me and always leaves food behind.
This morning, I divided a can of Friskies® Liver and Chicken into two bowls and set them out there. Lydia approached while Murphy watched and Lydia ate a few bites out of the bowl on our left, then returned to her house. Murphy then approached and ate the rest of the food in the bowl on the left and all of the food in the bowl on the right. Maybe whoever was feeding Murphy before hasn't been feeding Murphy lately.