Properly explain indeterminate forms to me

I mean, they're not wrong.

But you aren't either.

0/0 is undefined.

That said, there are algebraic expressions involving limits that look like 0/0 or similar forms during the limit  evaluation process.

So that's meaningful if you are resolving limits, but not so much  otherwise.
My reading is that the difference between "indeterminate form" and "undefined" is nuanced and maybe even picky. Both mean that an expression has no single well-defined value. Perhaps "undefined" means that no value could ever work, and "indeterminate form" means that more than one value could work. In that case 1/0 would be undefined, and 0/0 would be indeterminate.
Indeterminate forms means you can't determine the limit, so you need to try harder. A simple example would be x^2 / x
Limit as x approaches 0 from the positive side.
Its indeterminate, its 0/0, meaning i gotta try something else instead of just plugging 0. In this case you just need to simplify and get that it equals x, then the limit is 0.
Im pretty sure the reason we have them is because people showed you can get different answers from the same kind of limits. For example 1^infinity is indeterminate because its the same kind of number as e. There may be times where 1^infinity isn't e, which is why 1^infinity can't be determined to 1 or e so easily.
Neither intuition is way off, but both statements are wrong. The indeterminate form is NOT "0/0", it's

"lim\_{x -> a} f(x)/g(x), where f(x) and g(x) both go to 0"

Or the other versions. It's really a big official thing. It's a limit situation that's ambiguous because of competing extremes that merits extra investigation. And the "0/0" is not "archaic" it's a *cutesy shorthand* that you put *off to the side* as a reminder of the situation and an indication of why you're going to apply L'Hopital or whatever. It's for communication; not a real thing you put in your work.
0?/0? and 0?\^0? are both indeterminate and you need to do more work
0/0 0\^0 and 0?/0 are all undefined
0\^0? is 0
0?\^0 is 1
I'm using 0? to refer to a function that is approaching zero in some way.