I am reading through Peters-Steenbrink Mixed Hodge structures and I am having some trouble understanding what definitions of cohomology and hypercohomology are getting used. Let $U$ be a complex manifold. Its cohomology is (I think) by definition the cohomology of the constant sheaf $\underline{\mathbb{C}}_U$, which by definition is computed by taking an injective resolution, applying the global sections functor component-wise to the resolution and taking cohomology. This is the same as seeing $\underline{\mathbb{C}}_U$ as a complex concentrated in degree zero and taking hypercohomology:
$$ H^k(U;\mathbb{C}) := H^k(U;\underline{\mathbb{C}}_U) := \mathbb{H}^k(U;\underline{\mathbb{C}}_U) := H^k \circ R\Gamma(U,-)\ (\cdots \to 0 \to \underline{\mathbb{C}}_U \to 0 \to \ldots). $$
On the other hand, we can have some good compactification $j: U \to X$ and we want to work with sheaves on $X$. In this case I would notice that $\Gamma(U,-) = \Gamma(X,-) \circ j_*$ and hence
$$ H^k(U;\mathbb{C}) = H^k \circ R\Gamma(U,-) (\underline{\mathbb{C}}_U) = H^k \circ R\Gamma(X,-) \circ Rj_* (\underline{\mathbb{C}}_U) = \mathbb{H}^k(X; Rj_*(\underline{\mathbb{C}}_U)). $$
Since the holomorphic de Rham complex $\Omega^\bullet_U$ is a resolution of $\underline{\mathbb{C}}_U$, they are equal in the derived category, and hence
$$ H^k(U; \mathbb{C}) = \mathbb{H}^k(X; Rj_*\Omega^\bullet_u) $$
However this is not what I understand from the book (see for instance Proposition 4.3). Instead, they seem to take for granted that the cohomology of $U$ is the hypercohomology of $j_*\Omega^\bullet_U$, which I understand to be the component-wise application of the functor $j_*$ to the complex $\Omega^\bullet_U$.
What am I missing?
EDIT: I guess that I can rephrase my question as follows. Which of the following is true:
$$ \mathbb{H}^k(U, \mathcal{F}^\bullet) = \mathbb{H}^k(X, j_*\mathcal{F}^\bullet) \qquad \text{or} \qquad \mathbb{H}^k(U, \mathcal{F}^\bullet) = \mathbb{H}^k(X, Rj_*\mathcal{F}^\bullet)? $$
The argument above seems to tell me that the second one is true, but the book seems to imply that the first one is true.
If $j$ is an open embedding, $j_*$ is exact, and $Rj_* = j_*$.