Why NP is not closed under Turing reduction
The reason that $A$ is not defined as in NP is that we use a finer notion for NP-completeness, which is mapping (Karp) reductions.

Intuitively, in a Karp reduction, we not only want the problem $A$ to be solvable using an oracle for $B$, but we also require that this oracle is only used once, and only as the very last operation.

Why this notion and not Turing reductions? First - it works well, and it provides a finer distinction in complexity classes (more on that in a bit). Second, it captures the intuition that if you could solve $B$, then given input for $A$ you can convert it to input for $B$, rather than just use $B$ in some complex and clever manner.  So in a way, it makes "very little use" of $B$. This is all philosophical, of course.

In practice, the main benefit of Karp reduction is that it differentiates $coNP$ from $NP$. Indeed, you can show a Turing reduction from $SAT$ to $\overline{SAT}$ (just call the oracle and flip the answer). So you cannot have the notion of coNP-complete v.s. NP-complete if you use Turing reductions.

You can read the discussion here for a more elaborate answer.