I think your research has been misdirected.
Let us read a simple definition of at from Merriam-Webster:
used to indicate the person or thing toward which an action, motion, or feeling is directed or aimed
I make the assumption that shot and fire mean to discharge a weapon, like a pistol.
Now we consider the examples.
- He shot at the sparrow.
- He shot the sparrow.
1 means that he shot towards the sparrow, or in the direction of the sparrow. It does not necessarily imply that the sparrow was injured, killed, or even hit by the bullet. 2 implies that the sparrow was actually hit.
- He fired at the sparrow.
- He fired the sparrow.
3 is similar in meaning to 1. 4 does not sound right (to me at least) under the aforementioned assumption. One usually fires a weapon, a bullet, a rocket, etc., and not the target. So 4 does not sound like anyone shot or shot at the bird.
Since 4 doesn't make sense in terms of shooting, then, to me, the first definition that comes to mind for 4 is
- : to dismiss from employment
<He was fired from his job.>
So the sparrow was dismissed from his job. This is silly. :D
The second meaning that comes to mind is that the bird was roasted. In either case, the meaning of shooting does not come to mind.