I find the claim that in quantum state teleportation "No information was teleported" to be pretty much flat-out wrong. This much is true,
you interact information with an entangled system, get information back, and use that with the other part of the system to retrieve information,
but the information that Alice gets back from interacting with and measuring the system she intends to teleport does not give her any information about that state - and, even if Alice tried, she would be completely unable to obtain enough information about the state to transmit it classically.
The core point of the issue is that quantum teleportation involves the transport of (i) a quantum state, and moreover, (ii) the quantum state of a single instance of the given system.
If the system were classical, then Alice could just measure the system, transmit the results, and Bob would use those as instructions to reinstate the system, but quantum states don't work like that: given a single instance of the system, there is provably no way to fully characterize its state. If you have an ensemble of copies, you can perform quantum state tomography on them, but if you have a single qubit, say, there are a bunch of non-commuting relevant observables, and measuring one of them will collapse the state and destroy all the information about what would have happened if you measured the others.
When seen like that, it looks downright impossible that any protocol short of physical transport of the system will be able to transmit that full quantum state to Bob, but that is precisely what quantum state teleportation achieves. In that protocol, Alice actively relinquishes her chance to make a measurement that would give her information about the state, and instead she performs an entangled Bell-basis measurement with a second particle (itself entangled with a counterpart controlled by Bob) the results of which describe the type of correlation that would be induced on the 1-2 pair (were the second particle not entangled; since it is, the results describe how the state to be teleported has been encoded in Bob's particle) but they yield no information at all about what that state was.
This is what's transmitted in the two classical bits you mention,
A two q-bit system "teleporting" one bit of information requires two bits to recover the information, that's exactly 4 possibilities,
$-$ yes, four possible encodings of the original state into Bob's system. However, your assertion that these two bits of information are somehow
the exact amount of possibilities in the system
is flat-out wrong. The information that Bob acquires in the protocol isn't one bit and it isn't two, it is one qubit, and that is a special kind of information all its own, which just isn't comparable to bits or measurable in those units. (Indeed, if you want to make such a case, then a qubit contains an unbounded amount of information, and thus there is a (fairly useless) sense in which the protocol transmits an infinite (or at least arbitrarily large) amount of information.)
