I want to strengthen Alexey's answer, and claim that the reason is that the first definition suffers from technical difficulties, and not just that the second (standard) way is more natural.
Alexy's point is that the first approach, i.e.:
$M \models \forall x . \phi \iff$ for all $d \in M$: $M \models \phi[x\mapsto d]$
mixes syntax and semantics.
For example, let's take Alexey's example:
${(0,\infty)} \models x > 2$
Then in order to show that, one of the things we have to show is:
$(0,\infty) \models \pi > 2$
The entity $\pi > 2$ is not a formula, unless our language includes the symbol $\pi$, that is interpreted in the model $M$ as the mathematical constant $\pi \approx 3.141\ldots$.
A more extreme case would be to show that $M\models\sqrt[15]{15,000,000} > 2$, and again, the right hand side is a valid formula only if our language contains a binary radical symbol $\sqrt{}$, that is interpreted as the radical, and number constants $15$ and $15,000,000$.
To ram the point home, consider what happens when the model we present has a more complicated structure. For example, instead of taking real numbers, take Dedekind cuts (a particular implementation of the real numbers).
Then the elements of your model are not just "numbers". They are pairs of sets of rational numbers $(A,B)$ that form a Dedkind cut.
Now, look at the object $({q \in \mathbb Q | q < 0 \vee q^2 < 5}, {q \in \mathbb Q | 0 \leq q \wedge q^2 > 5}) > 2$" (which is what we get when we "substitute" the Dedekind cut describing $\sqrt{5}$ in the formula $x > 2$.
What is this object? It's not a formula --- it has sets, and pairs and who knows what in it. It's potentially infinite.
So in order for this approach to work well, you need to extend your notion of "formula" to include such mixed entities of semantic and syntactic objects. Then you need to define operations such as substitutions on them. But now substitutions would no longer be syntactic functions: $[ x \mapsto t]: Terms \to Terms$. They would be operations on very very large collections of these generalised, semantically mixed terms.
It's possible you will be able to overcome these technicalities, but I guess you will have to work very hard.
The standard approach keeps the distinction between syntax and semantics. What we change is the valuation, a semantic entity, and keep formulae syntactic.