... But hey, I'm open to learning the old lingo. It wouldn't hurt. It's just that most of the people I interact with, primarily online, tend to use the newer terminology, so we understand each other perfectly.
I think Ansel Adam's original Zone System, complete with "New Lingo," was introduced in his five Basic Photo series books, first published in 1948. The concepts were around, complete with the evolving "new lingo" for decades before. Adams, along with Fred Archer, developed the Zone System (itself a simplification and practical application of photographic science) to make things easier for practicing photographers. They invented a method that tied development time to the subject contrast range and based exposure on the lower values in the scene, thus guarding against underexposure. The adage, "expose for the shadows and develop for the highlights," however, had been around long before that.
It's actually "pushing" and "pulling" that are the newer terms. They are, simply, a more incomplete and inexact simplification of more complicated and meticulous practices based in scientific research, like the Zone System (or if you like, a dumbed-down even-dumber version of photographic theory for those who didn't want or need to understand the principles involved more completely).
The concepts of adjusting negative contrast by changing development time is key to understanding what's going on here. More development = more overall negative contrast and vice versa. Understand this, and the fact that a film's speed is determined at manufacture (not really flexible at all, as many would like to think) and you'll have a start.
Now, when film is underexposed, the lower values (shadows) just aren't recorded on the film (remember, we really can't change the film's speed!) and, as a result, the mid-tones and highlights that do get recorded end up being recorded as shadows and mid-tones. A scene with a seven-stop brightness range that is underexposed by two stops (i.e., rating your film two stops faster than it really is) ends up losing the bottom two of those seven steps. Step 3 is now the darkest shadow (which should have been a mid-tone) and step 7, which should be white, ends up being step 5, a middle gray.
What to do? Develop more so that we can raise that step 7 exposure to a density similar to what it would have been had the film been exposed and developed normally. This ends up giving you a negative with only 5 steps, stretched out between black and white, but maybe an acceptable image, especially if there were no other way to get the shot in a low-light situation. This began to be called "pushing" sometime in the mid-20th century (or maybe earlier). The term, however, is just an amateurish shorthand for overdeveloping in order to save an underexposed image. That underexposure might be intentional, as in low-light situations where there is no other choice, or not.
Pulling is the opposite and usually just refers to developing less in contrasty situations. Whereas the Zone System and other exposure/development systems are rather precise about how much less to develop for contrasty scenes, "pulling" is a rather imprecise approach to get the overall negative contrast somewhere in the printable range without having to worry about metering the scene carefully having done a lot of tests to determine just how much less development is optimum for a particular scene. Again, it's newer and less precise than what came before.
For photographers just coming to film, the more recent and less-precise terms are perhaps more readily available and approachable. Do know, however, that there is a whole world of more-precise practice out there waiting to be discovered.
FWIW, many of us dislike the terms "push" and "pull," just because they are so inaccurate and express an imcomplete understanding of the basics of film photography. Manufacturers do tend to use them, however, just because their target consumers are often amateurs and beginners; they feel that more complex or technical descriptions might put-off people from using their products and that the simplification makes them more appealing. They are likely correct.
Best,
Doremus