In 1 they define highest weight representations of a Kac-Moody algebra. In 2 they define highest weight representations of an untwisted affine Kac Moody algebra (which can be written in the form $\hat g = g[\mathbb C(t,t^{-1})]\oplus \mathbb C \hat k\ \oplus \mathbb C L_0$, where $g$ is a semisimple Lie algebra).
I do not know how the second definition is a particular case of the second:
From 1:
Let $l$ be a Kac-Moody algebra with triangular deomposition $n_-\oplus \mathfrak h\oplus n_+$.
A (Kac-Moody) highest weight $l$-module with highest weight $\Lambda \in \mathfrak h^*$ is a $l$-module $V$ containing a nonzero vector $v\in V$ such that:
\begin{align}
&n_+\cdot v = 0 \\
&H\cdot v = \Lambda(H)v\quad (\forall H \in \mathfrak h) \\
&U(n_-)\cdot v = V
\end{align}
The vector $v$ is called a highest weight vector.
From 2:
I know that we can write $\hat \lambda$ in terms of $\lambda$, $\hat\lambda (\hat k)$ and $\hat\lambda(L_0)$, so (14.121) is not surprising except in the fact that we demand $\hat\lambda(L_0)=0$. Furthermore, the writer also writes (with no justification):
However, I do not see why one is free to perform this redefinition...
New Information: In Fuchs' book on affine Lie algebras and quantum groups, they give the following definition for highest weight representation $R_\Lambda$ of an affine Kac-Moody algebra: $$ \begin{array}{l}{R_{\Lambda}\left(H_{0}^{i}\right) \cdot v_{\Lambda}=\bar{\Lambda}^{i} v_{\Lambda}} \\ {R_{\Lambda}(K) \cdot v_{\Lambda}=k v_{\Lambda}} \\ {R_{\Lambda}(D) \cdot v_{A}=n_{0} v_{\Lambda}}\end{array} $$ and they say the following:
1 Kac, V. G. (1990). Infinite-Dimensional Lie Algebras. Cam- bridge University Press, 3 edition.
2 Francesco, P., Mathieu, P., and Sénéchal, D. (2012). Conformal field theory. Springer Science & Business Media.