Confusing use of notation in expressing probability distribution
I'm trying to follow a text about Bayesian statistics, and the author is using the following notation to describe a random variable which has normal distribution:
p(x | µ, σ2) = (Gaussian density function here)
In a Bayesian text, this notation is confusing, since it makes me think about mean and variance as random variables, but they are not random variables. They are simply the parameters of the density function, and this is just using the conditional probability notation for expressing something else, namely saying that "x is distributed normally, with this mean and this variance"
This is not the case where you have p(a|b) = p(b|a).p(a)/p(b)
My question is, is there a unifying way of thinking about the first notation so that I don't have to distinguish between the case where I simply have description of a probability distribution, and the case where it is about conditional distribution? I could not get my head around the idea of interpreting mean and variance as variables on which the random variable is conditioned on. Is this really a different use of the same notation, or am I missing something here?
I hope I can describe my problem, and apologies if this is not clear enough. Your response would be much appreciated!