Ein Käfig ging einen Vogel suchen.
A cage went in search of a bird.
This one is translated identically in both editions.
The search is paradoxical. A bird is free, and if its freedom is considered a part of its essence, then a bird deprived of its freedom isn't the same bird anymore. I don't think the primary point here is that one may have an idea of some thing only to find that possessing that thing isn't the same as possessing that idea. Kafka is pointing out how the search for something pushes it away from you.
You want the bird, but why do you want the bird? Because it's free. So you catch a bird. Now it isn't free any more. How do you "have" a free bird?
The cage is formed around the bird, roughly in keeping with its dimensions, needs, and habits. Kafka may be saying that certain ideas are like this; they are attempts to trap something.
Searching, the cage becomes more like a bird; it would have to go where birds go, flying from branch to branch. So the cage may end its search by turning into a bird. Then again, it may turn into something entirely new, neither a bird nor a cage.
This means that the search does not always push the object away, but that when it doesn't, it also does not result in capture.