Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted αβ-statistical convergence of order γ, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted αβ-statistical convergence of order γ, which is an extension of αβ-statistical convergence of order γ. Our definition, with sk = 1, for all k, reduces to αβ-statistical convergence of order γ. Moreover, we use this definition of weighted αβ-statistical convergence of order γ, to prove Korovkin type approximation theorems via, weighted αβ-equistatistical convergence of order γ and weighted αβ-statistical uniform convergence of order γ, for bivariate functions on [0, ) [0, ). Also we prove Korovkin type approximation theorems via αβ-equistatistical convergence of order γ and αβ-statistical uniform convergence of order γ, for bivariate functions on [0, ) [0, ). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted αβ-equistatistical convergence of order γ is introduced and discussed.