Technically, you can only give a useful answer about the probability density of the random variable Z_k at the value z_k, conditioned on X_k = x_k. In the Bayesian interpretation of the Kalman filter, you never have an event "I measured z_k" (that event has probability 0, of course).
I agree that the probability notation is the issue here. Look at how wikipedia shows Bayes' rule for continuous random variables on both sides of the |. : https://en.wikipedia.org/wiki/Bayes%27_theorem#Random_variab... That's the kind of explicit and precise notation I would use to help someone understand the Kalman filter from a Bayesian perspective.
Once you use that definition of Bayes' rule, then you can substitute the definitions of the multivariate normal pdf, Do The Math, and derive the Kalman filter recursive updates.