probability ambiguous notation

Question

Part A)

I have the following notations :

1. $p(\mu, \sigma^2)$
2. $N(x)$
3. $p(x | \mu, \sigma^2)$

I would read them in the following manner:

1. probability function with 2 parameters, namely $\mu$ and $\sigma^2$
2. normal distribution of variable $x$
3. distribution of variable $x$ with parameters $\mu$ and $\sigma^2$

Part B)

If the above is correct, then please check my interpretation of :

1. $p(x,y)$
2. $p(x | y,a,b,c,d)$

my interpretation :

1. probabilty function of 2 independent variables
2. probability of variable $x$, given the probability for $y$ and parameters $a,b,c$ are known

Part C)

The follwoing notation is equal

$p(y∣x)$ is equal to $p_{Y∣X}(y∣x)$ is equal to $P(Y=y∣X=x)$

and the above notaitons in Part A and Part B can be rewritten in similar manner.

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I would conclude my understanding as follows:

$p(variable|parameters)$

where the $variable$ is the input to the function $p(x)$, and the $parameters$ are the parameters of $p(...)$ and as such, need to be known prior to the evaluation of $p(x)$