Making sure I understand this interaction right before I attempt to build a deck around it

Does this work as described?
I control Storm Cauldron and Price of Glory. If an opponent taps one or more lands for mana on my turn, as I am the active player, I decide how the Cauldron/Price triggers stack, and can put Cauldron's on the stack first, so the land will be destroyed by Price of Glory before it's returned to hand by Storm Cauldron. (Thus, when I cast Armageddon, only I can save lands by using the Cauldron to bounce them)

Same result, but for a slightly different reason. You choose how those two triggers stack because you control them, not because you're the active player. You always get to choose the order of your own triggered abilities.

Now, if abilities controlled by different players are being triggered simultaneously, first the active player puts all of his triggers on the stack in whatever order he chooses, then the nonactive player puts all of her triggers on top of the stack in whatever order she chooses.

In short, yes, your combo works.

As CommanderJim said, you are the only one who puts triggers you own onto the stack. It's just that the active player always puts theirs on first. The important thing to note is that this also applies if you tap your lands on your opponent's turn. As the active player, your opponent will put on any triggers they own onto the stack first, then you'll put on yours (choosing for whether the Storm Cauldron or Price of Glory resolves first).