Having been there (partly) & done that I think it's safe to say that the terminology can get right in the way of the concept if we're not careful (see discussions on bent/relaxed/straight arm elsewhere for examples)
Pushing with your armpit is a weird idea, not least 'cos there's no muscle there! But, using your serratus anterior to stretch/reach out/push toward the target is actually a way of getting your shoulder low & into mechanical line... so not pushing with your muscles.
You'll note the terms used by both camps figure in the description. I don't know if I'm describing the same end result, but I'd see both as trying for much the same thing described differently.