"She knows" has a definition: an extractor algorithm should exist that produces the witness $x$ while talking to the proving party in place of verifier. For zk-SNARK from Christian Lundkvist in the medium post you referenced, the protocol is non-interactive and the "proof of knowledge" is derived from the "knowledge of exponent assumption", not from the extractor algorithm. It seems there's a consensus that both definitions are good enough.
My point is, a non-interactive "proof of knowledge" is very different from an interactive one. It seems it was a design choice that zk-SNARK proofs should be verifiable by everyone without interaction.
There's a chance with a designated verifier (introduced in late 80s). As a setup, the potential verifier sends some hash $t$ to prover and runs this protocol to show he knows the witness $s$ for that additional hash: $H(s) = t$. The prover then runs a protocol to show his knowledge of either one of two preimages ($s$ or $x$).
This could mean a circuit representing equations:
z (1 - z) = 0 \\
H(zs + (1 - z)x) = zt + (1 - z)y
It follows only that a verifier that keeps his $s$ secret can be assured original statement $H(x) = y$ is true.
Make a protocol interactive again :)