Those koans were stories written by Mumon. They weren’t all based entirely on real events. Maybe Nansen gave a dharma talk where he said he would kill the cat, but to actually kill it would go against the precepts. Highly doubtful that actually happened.

The point of the koan isn’t surface level though, and if you expect to be able to extract a lesson from a koan by one simple reading, then you’re approaching koans wrong.

Mumon’s commentaries might help guide your understanding, and perhaps secondary scholarship like a translator’s notes or whatever. But unless you’re viewing it from inside the tradition, it’s likely to seem obscure/perplexing.

But unless you already understand what he means by “say a word of Zen,” and why Joshu placing a sandal on his head was taken as an acceptable answer, then this story will elude you no matter how you look at it.