The definition is, indeed, equivalent to a choice of complex structure on $TM \oplus \Bbb R^{k}$ for sufficiently large $k$ (roughly $k > dim(M)$). Roughly, this is because the tangent bundle and the normal bundle in $S^d$ sum to a trivial bundle, and so a stable complex structure on one can be transferred to a stable complex structure on the other. So this makes the definitions a matter of taste, but there are a couple of reasons why this definition might be viewed as "more basic".
One is that you don't have to talk about stabilization quite as much; you can just ensure that the ambient sphere has sufficiently high dimension.
Another is related to transferring a stable almost-complex structure to the boundary. The tangent bundle of the boundary $\partial W$ is a summand of the tangent bundle of $W$—specifically, $TW|_{\partial W} \cong T(\partial W) \oplus \Bbb R$. It is not hard, but somewhat irritating, to transfer the complex structure from the larger vector space to the smaller one, especially verifying that it is independent of choices made. By contrast, the normal bundle satisfies $\nu W|_{\partial W} \oplus \Bbb R \cong \nu(\partial W)$, and so to get a complex structure on $\nu(\partial W)$ we just have to throw on an extra copy of $\Bbb R$.
But probably most importantly, there is the cobordism classification. Ultimately, to connect homotopy theory to manifolds, you're going to start with a map $f: S^{m+2n} \to Thom(BU(n))$ that is transverse to the zero section, and take the inverse image of the zero section. This zero section $M$ is naturally going to have an identification of its normal bundle with a complex vector space—the pullback of the canonical bundle of $BU(n)$—because this map $f$ can be used to identify them. This makes the definition of a stably almost-complex structure better geared to how it connects to Thom's classification mechanism.
