Is the following definintion of Discrete Random Variable given in the book of Joe Blitzstein correct? The following is the definition given in the book:

A random variable $X$ is said to be discrete if there is a finite list
  of values $a_1, a_2,\ldots, a_n$ or an infinite list of values $ a_1, a_2,\ldots$ such that $ \ P(X = a_j \ \text{for some} \ j ) = 1 $.

Why should the probability of a random variable taking a value be one?
For example: Let a fair coin be tossed twice. Let $ X $ be the no. of heads. There can only be $P(X = 0) = P(X = 2) = \frac{1}{4}$ and $P(X = 1) = \frac{1}{2} $. There is no $k \in \mathbb{N}$ such that $P(X = k) = 1$. So does that mean from the above definition that $X$ is not a discrete r.v. ? 
 A: A good way to remedy your problem with the definition may be to think of the statement $\ P(X = a_j \ \text{for some} \ j ) = 1$ in "words" and apply this to your example. 
Suppose we define our list of outcomes (the $a_i$) by $a_1 = 0$, $a_2=1$, $a_3 = 2$ which is all possible numbers of heads. What the statement $\ P(X = a_j \ \text{for some} \ j ) = 1$ means in words is that there is a probability of 1 that our random variable $X$ will take on some value in our list. When flipping the coin twice we notice that our outcome will always be a value of either $0,1,$ or $2$ heads and so indeed $\ P(X = a_j \ \text{for some} \ j ) = 1$ holds as we always (with probability 1) take on some value in the list (either 0,1, or 2).
In short, what the author is getting at is that evaluation/observation of our random variable always lies in the range defined by the $a_i$. Using your example, we want to rule out the chance of suddenly seeing 18 heads after two rolls . 
Finally, I think the comment by @BruceET does a good job of explaining the logic you don't want to use here!
