Balanced binary tree and $\log(n)$ searching time

Question

Standard definitions and theorems say:
"A balanced binary tree is a binary tree structure in which the left and right subtrees of every node differ in height by no more than $1$" (Wiki) "A balanced binary tree needs $O(\log n)$ search time" (dito).
Now, this clearly doesn't look like "iff" to me - if your tree is slightly damaged, say, you have a standard fully balanced static binary tree with keys from $1$ to $2^n-1$ and now insert keys $2^n,2^n+1$ without reordering, so there, it's unbalanced now but still searchs in $O(\log n)$. Is there a sharper definition of balance known that is iff?