In Nim, players must remove objects from exactly $1$ heap, and the winning strategy involves converting all heap sizes to base $2$, and removing objects to manipulate to $0$ the digital sum in base $2$ of those converted heap sizes.
In Nim$_k$, players must remove objects from between $1$ and $k$ heaps inclusive, and the winning strategy involves converting all heap sizes to base $2$, and removing objects to manipulate to $0$ the digital sum in base $k + 1$ of those converted heap sizes. Nim, then, would be Nim$_1$ in this generalization.
It is interesting that altering $k$ in the rules affects the base of the digital sum but not the base of the heap sizes in the winning strategy; the apparent "$2$-ness" of Nim in its strategy is only partially explained by the generalization, and it actually turns out to be "$1$-ness" in its rules (as reflected in the naming convention Nim$_1$).
Is there some other rules parameter $j$ which could be altered such that the winning strategy would involve converting all heap sizes to some base $f(j)$, and removing objects to manipulate to $0$ the digital sum in base $k + 1$ of those converted heap sizes? A tempting candidate for $j$ is the number of players (perhaps suggesting that $f(j) = j$), but I'm skeptical of this because I've read that combinatorial games with more than $2$ players are hard to solve due to coalitions, and this strategy is simple. Other candidates may be the minimum number of objects removed per turn, or the minimum number of objects removed per heap per turn (either perhaps suggesting that $f(j) = j + 1$).
What English rules would produce such a generalization to Nim$_{j,\ k}$? Would there be different games corresponding to the $f(j) \leq k + 1$ and $f(j) > k + 1$ cases?