Let X be a random variable.
Consider X ~ F. It can be read as X has distribution F.
What is distribution referring here? Consider the following interpretations
1) If X is continuous, then F is a probability density function and if X is discrete then F is a probability mass function.
2) F is a cumulative distribution function.
Which of the above is correct? If not, what is the distribution the notation referring to?
Suppose that $X$ is a real-valued random variable. There are some probability textbooks which use the term "distribution of $X$" to refer to the cumulative distribution function (CDF) of $X$. In these books, the expression $X \sim F$ often means that $F$ is the CDF of $X$. I suspect this is the case in your example.
However, I think the most standard definition of "the distribution of $X$" is the probability measure $\mu$ on $\mathbb R$ induced by $X$: $$ \mu(A) = P(X \in A) $$ for any measurable set $A \subset \mathbb R$. For example, this definition is used in Folland.
In most places where I see the $X \sim \square$ notation, my answer would be "neither." The thing in the $\square$ is neither a pdf nor a cdf.
I often see probability distributions specified in a format like this:
$$ X \sim N(0,1).$$
This says that the random variable $X$ has a standard normal distribution (a normal distribution with mean zero and variance $1$).
I do not recall ever seeing $N(0,1)$ used as the name of a function. In my experience it is neither the pdf of the standard normal distribution nor the cdf of the standard normal distribution. It is merely the name of the distribution itself, which can be identified either as the continuous real-valued distribution with the pdf $$ f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} $$ or the real-valued distribution with the cdf $$ F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} dt. $$
