You maybe want to have a look at 

- P. Brosnan and R. Joshua. Comparison of motivic and simplicial operations in mod-$\ell$ motivic and étale cohomology. In: Feynman amplitudes, periods and motives, Contemporary Math. 648, 2015, 29-55. 

There are two sequences of cohomology operations in motivic and étale cohomology which can rightfully be called Steenrod operations. One (containing the Bockstein with the twist $\mathbb{Z}/4\mathbb{Z}(1)$) comes from a "geometric" model of the classifying space of the symmetric groups, the other one from a "simplicial" model. The first one is related to Voevodsky's motivic Steenrod algebra, the second one to the more classical Steenrod algebra (as in Denis Nardin's answer). 

The paper mentioned above provides a comparison between these sequences of cohomology operations (see Theorem 1.1 part iii for the étale cohomology) in terms of cup products with powers of a motivic Bott element in $H^0(\operatorname{Spec} k,\mathbb{Z}/\ell\mathbb{Z}(1))$. In the Bockstein case, the comparison theorem suggests that the two operations you defined are the same.