Idk where you teach, but I'm thankful you didn't teach me.
Let me quizz you, how do you solve 2(2+2)^2? because acording to your linked picture, because brackets are leftmost you do them first. If I were to believe you:
(2*2+2*2)^2
(4+4)^2, = 64
but it's just simply incorrect.
2(4)^2, wow we're at a 2x^2
2*16 = 32
The thing that pisses me off most, is the fact that, yes. Terms exists, yes they have all sorts of properties. But they are not rules, they are properties. And they only apply when we have unknows and we're at the most simplified form. For example your last link, the dude told us that those terms get prio because they are terms!? There are no mention of term prio in the book. It just simply said that when we have a simplified expression like: 2x^2+3x+5 we call 2x^2 and 3x and 5 terms. And yes they get priority, not because we named them those, but because they are multiplications. These help us at functions the most. Where we can assume that the highest power takes the sign at infinity. Maybe if the numbers look right, we can guess where it'd switch sign.
I don't even want to waste energy proofreading this, or telling you the obvious that when we have a div. and a mult. and no x's there really is no point in using terms, as we just get a single number.
But again, I totally understand why someone would use this, it's easier. But it's not the rule still. That's why at some places this is the default. I forgot the name/keywords but if you read a calculator's manual there must be a chapter or something regarding this exact issue.
So yeah, use it. It's good. Especially if you teach physics. But please don't go around making up rules.
As for your sources, you still linked a blog post.