So let me start again. If $\mathcal M$ is a coherent locally free sheaf on $X$, then the total space of the associated vector bundle $M$ is the affine $X$-scheme $M =\mathrm{Spec} (\mathcal{Sym}_{O_X}(\mathcal M^{\vee}))\to X.$ (the cech means dual.) This is an affine morphism, hence quasi-projective. As $X$ has an ample sheaf, this implies that $M$ admits an open immersion into some $\mathbb P^n_X$ (EGA, II.5.3.3).
This immersion can be "explicitely" constructed as follows. Let $n\ge 1$ be big enough so that $\mathcal M^{\vee}(n)$ is generated by its global sections. Write $ O_X^m \twoheadrightarrow \mathcal M^{\vee}(n), \quad O_X^3\twoheadrightarrow O_X(n)$ (If $T_0, T_1, T_2$ is a basis of $H^0(X, O_X(1))$, then $O_X(n)$ is generated by $T_0^n, T_1^n, T_2^n$). Then $\mathrm{Spec} (\mathcal{Sym}_{O_X}(\mathcal M^{\vee})) \hookrightarrow \mathbb P(\mathcal M^{\vee}\oplus O_X)\simeq {\mathbb P}(\mathcal M^{\vee}(n)\oplus O_X(n)) \hookrightarrow {\mathbb P}(O_X^{m+3})={\mathbb P}^{m+2}\times X.$
When $\mathcal M=O_X(d)$, we can take $n=d$ and $m=1$. Hence an immersion $L \hookrightarrow {\mathbb P}^{3}\times X.$
Now to write the map in your chart: over $U_i$, the map is $(\lambda_i, [x_0, x_1, x_2])\mapsto ([\lambda_i x_i^d, x_0^d, x_1^d, x_2^d], [x_0, x_1, x_2]).$
Le glueing map on $U_i\cap U_j$ is $\lambda_i x_i^d=\lambda_j x_j^d$.
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Below is the embedding for the total space of $O_X(-d)$.
Let $e_1,\dots, e_n$ be a basis of $H^0(X, O_X(d))$. This means that the map $O_X^n\to O_X(d), \quad (f_1,\dots, f_n)\mapsto \sum_i e_if_i$ is surjective. So $L$ is a closed subvariety of $\mathbb A^n\times X$ and is therefore quasi-projective.
This holds for any quasi-projective variety $X$ and any line bundle $L$ on $X$ generated by its global sections.
EDIT Please forget the followings lines.
It can't be. Because otherwise you would have a non-constant morphism from $\mathbb P^3$ (your total spae has dimension $3$) to $\mathbb P^2$. But such a morphism doesn't exist (this is an exercice in Hartshorne: there is no non-constant morphism from $\mathbb P^n$ to $\mathbb P^m$ if $n>m$).