There are a trio of similar, sometimes confused, bases: the frog base, the lily base, and buttonhole flower base (named by conjecture). The difference being the completion of the petal fold and the orientation of the petal flap afterwards. The buttonhole flower base does not have the petal fold, the lily base has the petals pointed towards the points formed from the corner of the square, and the frog base has the petals pointed towards the point formed from the center of the square.
The frog base name dates back to at least the 1960s and 70s when it was documented in books published by Samuel Randlett (Art of Origami) and Robert Harbin (Secrets of Origami). A later book, The Magic of Origami, by Alice Gray, Kunihiko Kasahara and Lillian Oppenheimer, referred to this base as the frog or lily base, but attempted to promote the lily name because of the proliferation of frog models folded from other bases. However, this distinction did not seem to take hold.
The buttonhole flower base is a name based on the conjecture of Hans Dybkjær on the origami-l list in July of 2012 due to the fact that the buttonhole flower is the simplest traditional fold associated with the squashed flap but non-petal folded form of the preliminary base.
In the same discussion of bases, Michael LaFosse attributes the name of the lily base to a discussion with Lillian Oppenheimer where she made the flap orientation distinction between the lily and frog bases.