This doesn’t mean the input is “true” though. I can send some zkp that says I’m at least 18 or that I have at least $1M in my bank account, but it doesn’t mean that I actually am or do. Or am I missing something?
This setting is generally "proof of attribute" or "proof of group membership" although in practice it is most often more like "proof of possession of a credential". An interesting example demonstrated a few years ago is that you could prove that you possess a passport from a certain country without revealing anything else about your identity (as the passports are digitally signed by their issuing authorities using publicly-known keys). You could then have, for example, an online forum or poll that only allows participation of people with a certain credential, yet the forum or poll operator never learns the offline identities of the members or participants.
There are some logistical issues with this depending on the purpose for which the verifier is relying on the statement, including what happens if a prover submits the same credential twice, and what happens if a prover borrows a credential from someone else. In some settings this is OK or unlikely, while in other settings it might effectively blow up the whole application!
the nullifier can then be your cryptographic identity as a member of some group. without disclosing the actual member.
it will likely be some time before such structures see use.
imagine a physical meeting between 1000 people and they all exchange some random-seeming string (prepared ahead of time), then they join a special group chat where they know there can only be 1000 members and each one corresponds to someone who was present in the meeting. but unless they out themselves (or everyone else does) they will never know who is who. and yet the group chat can be entirely p2p and no one can cheat. and there could be spy cameras watching every exchange in the meeting and all the exchanged notes and it wouldn't matter.
that way that works is by using a ZKP to prove that a previously secret but now disclosed nullifier string is a cryptographic relation to one member of the sorted list of all the exchanged strings in the meeting. and the nullifier happens to be a public key hash.
A non-augmented ZKPP protects the password from eavesdroppers, but the server's verifier is a password-equivalent (though it isn't the password, just a one-way function of it).
A ZKPP is the simplest ZKP application, but others work similarly.
In other words: you're missing everything about ZKPs.
if you want to do it without a third party you end up with something like zcash. monero plans to follow suit (currently only the amounts in monero are zero knowledge).
Or you can make it for remote programming job. Create a zk proof of codes he has worked on. And apply to HR so hr doesnt have to waste time on due diligence or mess with nda to look at private code base, few devs have public code. Zk makes, cross institute as an online service more efficient.