A "pony" is ...
A "stage" is...
A "deco bottle" ...
Those definitions seem quite reasonable and common.
You gotta feel reasonably confident the pony is big enough to get you to the surface in one piece however. To do that, you should probably do some calculations (which incorporate several important assumptions) to see how much gas you need to surface.
I also advocate testing those assumptions and calculations. For example, surface from ???-meters, using your ??-cu bottle, and see how much air remains. Even add in some swimming/exercise to simulate a higher breathing rate.
I studiously avoid going into deco because I don't currently dive with a redundant gas supply.
[Part 1]
Personally, I carry a redundant and independent air supply even on 30ft (10-meter) dives. I understand some people can or have CESA from depth, but it's not a gamble I want to take. I treat my redundant supply as a sort of "health insurance" or "PTSD insurance." Effectively, no matter what goes wrong
(whether it's entanglements, regulator problems, lost fin, OOA because I'm a dummy, and so on ....), I always have proper gas planned in my primary, and a bonus completely independent air supply sufficient to safely surface.
I understand far less than half of divers do carry a redundant air supply, but what if you need air and you've lost visibility or your buddy swam away? Your profile says "instructor," would you be confident with your students sharing gas with you. What if a student is hyper-ventilating, do you have more than enough for the both of you?
[Part 2]
I'm far from the most strict diver you'll meet, I "self taught" sidemount after-all
(don't recommend, hah), but
I get a strong sense you're attempting full-on technical-diving while avoiding tech-diving classes.
I'm not a technical diver, but most of the diving you describe is right on the gray-line between recreation and technical diving, perhaps occasionally finning over that line. You're also asking a lot of questions which are probably answered in most standard technical-diving training. It depends on who is offering the course, but technical diving is often broken down into levels.
If you take "tech 1" (or equivalent), but try to stay out of deco that's fine, it's better to be a little over-prepared, than borderline unprepared. Even if I've taken "Advanced open water" or "nitrox" that doesn't mean I have to dive to 40-meters all the time, at night, with nitrox every dive ... it's just an option I have.