I'm confused as to why it auto changes the settings completely different in full auto to shutter auto. When the only thing logically that would be different is shutter speed. 1/125 seems excessive also for what you were shooting and at iso 400. Iso 100 and a slower shutter would give you better images.
Problem is....it's overcast...the ISO 100 value for 1/2 second at f/5.6 was ISO 100,. giving blur to the moving water with no ISO gain needed. The first shot needed a Low ISO gain value, to reach 400 ISO, but that pulled f/5.6 at 1/125 second and a slightly under-exposed rendering--much,much,muuuuch safer for hand-held, automatic shooting.
In a matrix metering scenario, the Nikon's meter read the entire scene, computed the brightness range, the color values, and for the AUTO shot, delivered a good, solid, basic exposure of f/5.6 at 1/125 second at ISO 400...exactly the kind of compromise the Nikon engineers decided was about right, based on the 100,000+ actual scenes that are used to determine the Nikon matrix metering's exposure-making decisions.
When the photographer biases things to a very slow, non hand-holdable speed of 1/2 second in daylight, the camera's going to expose with that user-selected bias as a priority.