Cross-entropy is not the KL divergence. There is an additional term in cross-entropy which is the entropy of the data distribution (i.e., independent of the model). So, you're right in that minimizing one is equivalent to minimizing the other.
The first slide makes the disclaimer that one is not "able to identify if an event was due to climate change", only likelihoods of certain events occurring under various conditions. People might treat those two things the same due to our mental heuristics, but they are not.
https://stats.stackexchange.com/questions/357963/what-is-the...