Mel Krieger’s concept of a “pulling” stroke as opposed to a “pushing” one wasn’t intended as a way of differentiating two different casting styles, it was coined to describe any good casting stroke whatever the style used. So you should “pull through” when using an “elbow up-down” style, as much as when using an “elbow backward-forward” one.
Pulling was the term Mel chose to convey the idea of leaving for the end of the stroke as much of the rod rotation as possible. In summary (and using two other Mel’s concepts) “pulling” isn’t a style, it is substance; “pushing” isn’t a style, it is a fault.
The gif above shows the paradox posed by the interpretation of “pulling” as just an stylistic issue: Mel himself is showing what he meant by “pulling through” while performing a cast that some instructors would still define as “pushing”.
Better to let Mel explain himself about the “pulling” concept, and what the gif above is showing (bold is mine):
In the pull through casting stroke, the casting hand precedes the rod tip through most of the casting stroke and the turnover and stop take place only at the end of the casting stroke… Lay out 70 or so feet of fly line on a lawn behind you, fly rod pointing to the fly, and throw a javelin, turning the rod over only at the very end of the throw. You may be pleasantly surprised with this extreme pull through casting motion.