B-tree branching factor boundaries
More exactly, a B tree has an $m$ value such that the number of children is between $\lceil\frac m2\rceil$ and $m$ children. That differs from your numbers for odd $m$. For $m=3$ the bounds are $2$ and $3$, which is the 2-3-tree you refer to.

As far as I know (I tried this several years ago) these numbers are the smallest interval for which the split and merge operations can work. You can try yourself. If a node has $m$ children, I can add $1$ and the node must be halved. Thus this satisfy the $\lceil\frac m2\rceil$ bound? Check for $m$ odd and even separately.