In section 2.2 of Cache-Oblivious B-Trees, Strongly Weight-Balanced Search Trees are defined as:
For some constant $d$, every node $v$ at height $h$ has $\Theta(d^h)$ descendants.
Search trees that satisfy Properties 1 and 2 include weight-balanced B-trees, deterministic skip lists, and skip lists in the expected sense.
I can't figure out why deterministic skip lists have this property. The original paper on deterministic skip-lists notes that
As we see from Fig. 1, there exists a one-to-one correspondence between 1-2 skip lists and 2-3 trees.
It seems to me, however, that 2-3 trees are not strongly weight-balanced, since a node at height $h$ can have anywhere from $2^h$ to $3^h$ descendants.