I am confused whether the marginal distribution of a variable refers to its entire distribution or only the probability of a certain value which it assumes. My textbook has the following definition for a marginal distribution:
When given a joint distribution for $X$ and $Y$, the distribution $\mathbb{P}[X=a]$ for $X$ is called the marginal distribution for $X$, and can be found by "summing" over the values of $Y$. That is, $$ \mathbb{P}[X=a]=\sum_{b \in \mathscr{B}} \mathbb{P}[X=a, Y=b] $$ where $\mathscr{B}$ is the set of all possible values taken by $Y$.
Here, it seems we only get the probability of a single value which $X$ assumes, $a$. However, my textbook also claims that
The marginal distribution for $X_i$ is simply the distribution for $X_i$ and can be obtained by summing over all the possible values of the other variables.
This makes me think that the marginal distribution is the entire distribution of a variable, not only at a single point. So which of these views is correct, and how am I misinterpreting the information I am given?
Your doubts are justified. The different values of $a$ are missing in the definition of the marginal distribution. The right definition would be
$$ \mathbb{P}[X=a]=\begin{cases}\sum_{b \in \mathscr{B}} \mathbb{P}[X=a, Y=b] \quad \color{blue}{\forall \ \ a \in\mathscr{A}} \\ 0, \textrm{elsewhere}\end{cases}$$ where $\mathscr{A}$,$\mathscr{B}$ are the sets of all possible values taken by $X,Y$, respectively. The blue part considers all possible values of $a$. Thus you receive the pmf of $X$.