Yes, you're absolutely right.
Your images are two good examples of the amount of light required by the 70-300mm especially at long focal lengths. That's why a lot of people shooting birds tend to shoot in Aperture mode as opposed to Auto.
By using auto, you're allowing the camera to dictate the aperture and shutter speed. The 1st image is good even though, I think that the 70-300mm typically is sharpest at f8. When the light, in the 2nd shot was reduced, the camera adjusted the aperture down to f6.3. Probably a wee bit too low (IMHO) to get as sharp an image.