CLRS and Knuth disagree in the definition of “complete k-ary tree”

In CLRS (as recently as the fourth edition), a "complete k-ary tree" is defined as a tree in which "all leaves have the same depth and all internal nodes have degree k".

Knuth (as well as nearly every other source I'm aware of) uses an entirely different definition: "a k-ary tree in which all levels are filled except possibly the last, and in the last level, all nodes (which must be leaves) are as far to the left as possible".

One consequence is that the Knuth definition allows leaves at different depths. For example, a tree with two nodes: one root and a child, is a complete binary tree according to Knuth, but not according to CLRS.

I emailed the authors of CLRS a few months ago before the fourth edition was released, and was told that they acknowledge the difference, but did not have space to mention it because they were up against a publisher-enforced page count.