I don't think there are any formal definitions for any of them. They're basically what they say they are.
Game diagrams can become quite detailed. It depends a lot on the maturity of the product and the detail it is specified with. On a mature project with large teams that is moving forward meticulously, the details might include precisely what should happen at every step, with descriptions of the UI elements, descriptions of each animation and approximate timing, descriptions of the audio, descriptions of the VFX, descriptions of the math programmers should be using, descriptions of what variations must be available, descriptions of what values need to be adjustable by designers, descriptions of what should be customized by the player, and descriptions of what the overall experience should feel like.
Screens can similarly become quite detailed. What does every button do? What should be on the screen's background? That's in addition to the overall UI look and feel for the UI elements, animations of buttons/sliders/highlights, audio for interaction, VFX on interactions, haptic feedback (rumble), and experience generally. Every button and slider indicating what it is for, every transition indicating where it goes next, with consideration about how to improve the experience, simplify the direct experience to hopefully remove all barriers, ranging from make advanced options available to advanced players on one end to having a big “Play Now” button as the one-click option instantly available at startup. Descriptions can become quite detailed in large, mature projects so that everyone involved can know what to expect, and to help identify ways to improve.