Two definitions of balanced binary trees
Definition 1. is also known as weight-balancedness¹ and definition 2. as height-balancedness.

Height-balancedness does not imply weight-balancedness; examples are both AVL- and Red-Black-Trees. See here and here for proofs, respectively.

Weight-balancedness does imply height-balancedness, though. This can be proven by showing the following stronger fact by induction (over height): a weight-balanced tree is complete on all levels but the deepest². The essential argument in the inductive step is that the subtrees can't have a height difference of more than one because -- both having the claimed property by induction hypothesis -- they would then not be weight-balanced.




The article gives a different, more general definition.
In other words, such a tree of height $k$ without the leaves at level $k$ is a perfect tree of height $k-1$.