Definition of Lebesgue measurable functions- Why Borel sets?
Elaborating on my comments above:
One reason we should worry about the $(\mathcal{M}_{Leb}, \mathcal{M}_{Leb})$-approach is that with respect to $\mathcal{M}_{Leb}$, null sets are "too good." Specifically, no subset of a null-set is non-measurable. This means that no bijection $b$ between a positive-measure set $S$ and a null set $N$ can be $(\mathcal{M}_{Leb},\mathcal{M}_{Leb})$-measurable: consider $b[A]$ for $A\subseteq S$ non-measurable. Since there are continuous bijections between some positive-measure sets and some null sets, continuous functions won't in general be $(\mathcal{M}_{Leb},\mathcal{M}_{Leb})$-measurable.
More abstractly, we're seeing here that $\mathcal{M}_{Leb}$ and $\mathcal{M}_{Bor}$ are fundamentally different types of object:
$\mathcal{M}_{Leb}$ involves more than just the topology of $\mathbb{R}$. Let $C$ be the usual Cantor set and $F$ the fat Cantor set. Then there is an autohomeomorphism $h$ of $\mathbb{R}$ with $h[C]=F$. Consequently, by the above reasoning membership in $\mathcal{M}_{Leb}$ is not "ambient-isomorphism-invariant."
By contrast, $\mathcal{M}_{Bor}$ is purely topological: if $B\in\mathcal{M}_{Bor}$ and $h$ is an autohomeomorphism of $\mathbb{R}$ then $h[B]$ is also Borel.
Note that we have to be very careful here: the continuous image of a Borel set is not in general Borel! (The continuous preimage of a Borel set is Borel, however, and that's what's at work here.)
Also note that I'm not saying that $\mathcal{M}_{Bor}$ constitutes a topology on $\mathbb{R}$ - it doesn't, since all singletons are Borel but not all unions of singletons are Borel. I'm just saying that it is "reducible to" the topology in some sense. Indeed, every topological space comes with a notion of "Borel-ness" (and variations!) while nothing of the sort is true as far as measurability goes.
So in general we should expect Borel-ness to play well with topological concepts, but be suspicious of Lebesgue measurability's topological behavior. And the situation with continuous functions is a good example of one such discrepancy.
Finally, there's another vague theme here besides "topological nature:" structural rigidity. In a sense, some null sets are "measurable by accident" (namely, those which are very topologically different from any positive-measure measurable set). By contrast, nothing is "Borel by accident." This suggests that the class of Borel sets has much better structure overall than the class of Lebesgue measurable sets. This is true, and thinking along these lines takes us into the realm of descriptive set theory, but that's a far ways off. Still, it's worth pointing out here since the "flavor" at least is detectable already.