If $\rightarrow$ in $P \rightarrow Q$ is to be considered a logical connective, then he seems above to indicate neither more nor less than that it's truth value is "true" if and only if the truth value of $P$ (in each set up, as he says) is at most the truth value of $Q$. In other words, $\top \vdash P \rightarrow Q$ if and only if $P \vdash Q$. This doesn't really indicate what the truth value would be when the truth value of $P$ is (in some set up) not non-strictly below that of $Q$. So why doesn't he just use $\vdash$? I don't know.
Especially when conjunction and disjunction mean infimum and supremum respectively (when the entailment relation $\vdash$ is considered as giving a preorder), it tends to be useful to define the implication so as to make a Heyting prealgebra, i.e., so that $P \rightarrow Q$ is a largest formula whose conjunction with $P$ entails $Q$ . We know from what Belnap says that the truth value of $P \rightarrow Q$ must be $true$ when $P$ entails $Q$ (nine entries in the truth table), as is the case with logics having an implication that makes a Heyting prealgebra. What would demanding that implication makes a Heyting prealgebra mean in the other cases? Supposing $P$ and $Q$ to have truth values and that the truth value of $Q$ is not false and that $P$ does not entail $Q$ (four cases), this would cause the truth value of $P \rightarrow Q$ to be the truth value of $Q$. As for the remaining three possibilities, the truth value of $none \rightarrow false$ would be $both$ and the truth value of $both \rightarrow false$ would be $none$, and the truth value of $true \rightarrow false$ would be $false$. But it sounds like Belnap doesn't even define $\rightarrow$ as a connective.
Even in classical logic, entailment (denoted by $\vdash$) means something different from implication (denoted by $\rightarrow$). For instance, $(\top \rightarrow x \neq 0) \vee (\top \rightarrow x = 0)$ is a tautology (thus, true), but the metastatement "$\top \vdash x \neq 0$ or $\top \vdash x = 0$" is false; one could argue that the problem with this example results from not distinguishing the operator $\vee$ from the meta disjunction operator "or", but since in oral discourse the same words are used, that is a problem! At best conflating implication and entailment would seem to make it necessary to split hairs many other places--a poor choice that quite possibly it seems to me can't even work coherently or at all elegantly even when hairs are carefully split elsewhere.