In your terminology, ordinary first order logic would be $L_{\omega,\omega}^\omega$.
Consider the case of $L_{\omega,2}^\omega$. Here, we close the atomic formulas under finite conjunctions, negation and quantifiers over one variable. (By induction, this also gives rise to all of the assertions of first order logic, since any finite block of quantifiers can be thought of as happening on at a time.) In particular, any finite quantifier-free expression of first order logic is in this logic.
The logic $L_{\omega,\infty}^2$, in contrast, does not include any quantifier-free formulas with more than one variable.
So they are not the same.