Notation for conditional within a probability distribution

I'm reading an excellent tutorial on variational autoencoders by Carl Doersch. However, he uses the following notation to define the generative distribution:

$$P(X|z;\theta) = N(X|f(z;\theta), \sigma^{2} * I)$$

I interpret this to mean that our reconstruction of the observed variable $$X$$ is a normal distribution with $$\mu$$ defined by some function $$f(z;\theta)$$ and covariance $$\sigma^{2}$$.

And he uses this notation to describe the latent deep Gaussian distribution on $$Z$$:

$$P(Z) = N(Z|0,I)$$

Which I interpret to mean that

the latent variable Z is drawn from a (potentially multi-variate) normal distribution with $$\mu$$ vector of 0 and covariance $$I$$.

I don't understand what the difference between the above and simply writing

$$P(Z) = N(0,I)$$ and

$$P(X|z;\theta) = N(f(z; \theta), \sigma^{2} * I)$$.

What is the difference conceptually? Is there any, or is it just a matter of style?

That is the density notation in the paper. He reserves $$\mathcal{N}(0,I)$$ for referring to (normal) random variables (i.e. the distribution), and uses $$\mathcal{N}(X|0,I)$$ to refer to densities. For example, if $$Z$$ is distributed normally with zero-mean, identity covariance, we'd write $$Z\sim\mathcal{N}(0,I)$$, (as is in Page 11). When we're talking about $$p_Z(z)$$, that is something like $$\mathcal{N}(z|0,I)$$ in the paper. However, notice that Eq. 12, I believe mistakenly, doesn't follow this convention. This notation is also prevalent in the original VAE paper of Kingma.