This is a fascinating question! I really like your relation.

Here is some small progress. (I am hopeful that more definitive answers will appear later).

First, you didn't mention it, but for definiteness let's record
the fact that $\leq_L$ is included in $\leq_{L,end}$. That is, if
$W\models A\leq_L B$, then also $A\leq_{L,end}B$ with respect to
$W$. The reason is that if $W\subset W'$ is an end-extension with
$B\in L^{W'}$, then since the ordinals of $W'$ agree with the
ordinals of $W$ up to the height of $W$, it follows that when $W'$
constructs its version of $L[B]$, it will see that $A$ is added at
the same stage that puts $A\in L[B]$ from the perspective of $W$.

Second, someone might worry that whenever you have an
end-extension $W\subset W'$ with $B\in L^{W'}$, even though
$B\notin L^W$, then it might put all of $W$ into $L^{W'}$. In
other words, one might worry that $\leq_{L,end}$ collapses
everything to one equivalence class. But let me prove that this
isn't the case, even when one considers only well-founded
extensions.

To see this, suppose we have $L_\alpha\models\text{ZFC}$ and some larger countable $L_\beta\models\text{ZFC}$, with $L_\beta\models\alpha$ is countable. So inside $L_\beta$ there is an $L_\alpha$-generic Cohen real $B$. Now, let $A$ be an $L_\beta$-generic Cohen real, which is of course also $L_\alpha$-generic and indeed $L_\alpha[B]$-generic. Consider $W=L_\alpha[A,B]$. It is easy to check that this is end-extended by $W'=L_\beta[A,B]=L_\beta[A]$. Since $B\in L_\beta^{W'}=L_\beta$ and $A\notin L^{W'}$, it follows that $A\not\leq_{L,end} B$ with respect to $W$. And so the relation does not collapse to a single equivalence class.

It seems to me that one will be able to use this kind of reasoning to
show that other interesting things happen.

**Update.** Here is a negative answer to your final question for the version of your relation where one allows only well-founded models $W$ and $W'$.
That is, if $W$ is a countable transitive model of ZFC and $W\subset L$ (but not necessarily $W\models V=L$), define $A\leq_{L,end,wf}B$ for $A,B\in W$ just in case whenever $W'$ is a transitive end-extension of $W$ and $B\in L^{W'}$, then also $A\in L^{W'}$. We should probably assume that there are unboundedly many countable $L_\alpha$ modeling ZFC for this to be robust.

**Theorem.** In this context, $\leq_{L,end,wf}$ can differ from $\leq_L$ in the sense of $W$.

Proof. Let $L_\alpha$ and $L_\beta$ be as above, and choose $A$ to be $L$-least in $L_\beta$ that is an $L_\alpha$-generic Cohen real, and let $B$ be any $L_\beta$-generic Cohen real with $B\in L$. Thus, $A,B$ are mutually $L_\alpha$-generic Cohen reals, and $W=L_\alpha[A,B]$ is a model of ZFC. By mutual genericity, $W$ thinks $A\not\leq_LB\not\leq_LA$. But meanwhile, any well-founded $W'$ extending $W$ for which $B\in L^{W'}$ must be at least $\beta$ in height, and so $A\in L^{W'}$ as well. So $A\leq_{L,end,wf}B$. So the orders are different. QED

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