Someone told me, depending who is watching, they just use the waterstream to push the cat from the tree.

Depends.
In the first place: No, it has nothing to do with the joke!
In the second place: It is the reason the cat did climb the tree. The title could not be "Poor Career Choice" if the cat had climbed the tree for are reason that is unrelented to the cats job!

Cats often climb on trees. They could climb down or just jump, but often, they act like they are afrait.
People notice them, and try to climb the tree and rescue the cat, or call the firefighters to come with their ladders.

In the end, happily climbing a tree, but then act afrait when they should go back is a typical cat thing! :)