It turns out that my question is all about linear algebra. As I mentioned in the chat, to find such a quadric surface, one needs to find a line $L'$ in $\mathbb P^4$, which lies in the normal direction of $L$ inside the cubic threefold $X$. Such a line is the image of a nonzero section $s$ in the normal bundle $N_{L|X}\cong \mathcal{O}\oplus \mathcal{O}$ and the quadric surface is determined by the family of disjoint lines $\cup_{t\in \mathbb C}ts(L)$. 

In other words, by linearizing the local equation of the Fano variety of lines $F$ of $X$ in the Grassmannian $Gr(2,5)$ (i.e., throw out higher-order terms in 6.14 in Clemens-Griffiths), we can determine that the equation of $L'_t=ts(L)$ is

 $$
    \begin{cases}
    tax_1+x_2=0,\\
    tbx_0-x_4=0,\\
    tax_0-tbx_1-x_3=0.
    \end{cases}\tag{1}\label{1}
    $$

with $(a,b)\in \mathbb P^1$. By standard linear algebra, we cancel out $t$ and find that the $\mathbb P^{1}$-family of quadric surfaces are given by 

 $$
    \begin{cases}
    \text{I}.~a^2x_4+b^2x_2-abx_3=0,\\
    \text{II}.~ax_2x_0+ax_3x_1-bx_2x_1=0,\\
    \text{III}. ~ax_4x_0-bx_4x_1-bx_3x_0=0.
    \end{cases}\tag{2}\label{2}
    $$

in $\mathbb P^1_{[a,b]}\times \mathbb P^4_{[x_0,...,x_4]}$, with one linear equation and two quadric equations. Note that the family is not a complete intersection: When $a\neq 0$, the equation $\text{III}$ is redundant while when $b\neq 0$, the equation $\text{II}$ is redundant. Moreover, the points $(a,b)=(1,0)$ and $(0,1)$ correspond to the two "obvious" quadric surfaces that mentioned in the question.

By the way, I finally fixed the typo in Clemens-Griffiths p.309, about the defining equation of the curve $B(\alpha_0,\alpha_1)$: the fourth equation should be ````u_5=\cdots=u_n=z_4=\cdots=z_n=0````. (The original paper has a ````z_2```` instead of ````z_4````, which is wrong.) It turns out that the 1-parameter family of lines in Clemens-Griffiths is exactly the family $(\ref{1})$ we defined and $(\ref{2})$ are the explicit equations for
the $\mathbb P^1$-family of quadric surfaces tangent to $X$ along $L$ mentioned in Lemma 6.18.