Are there animals which can see movements better than shapes or colors?
I think "movement-based vision" is perhaps a bit of a misnomer. Ultimately, all vision is contrast-based, depending on integrating signals from ON and OFF bipolar cells. It seems that this notion of so-called "movement-based vision" is derived from work on toad vision and feature detection. In this case, it's more the downstream processing that results in different behavior based on the shape of the stimulus and the direction that it moves. The simple sensing of visual stimuli does not seem to be motion-based. 

So, aside from evolutionary inference, we would have no way of knowing whether Tyranassaurus rex had such behavior. As for the chickens, it's possible but I would be skeptical of such a system discriminating between immobile food and hand pouring food. I would chalk that up to a simple behavior of associating seeing you with getting food.

One interesting note: many, many species have reflexive responses to moving visual stimuli. The first is called the optokinetic reflex, in which the eyes reflexively track contrast-based movement; and the second is the optomotor response, in which the head or even the whole body move in response to a moving stimulus. Thus it is safe to say that for many (most?) species, motion detection has a strong influence on the visual response...but again, I remain hesitant to use the phrase "motion-based vision".