[A few interesting comments to question made me understand that I need to address some more fundamental facts. (The standard literature is elusive.)]
What (if any) is the correct way to translate Hilbert's Basis Theorem into logic / model theory?
Hilbert Basis Theorem comes in two forms.
The modern form
If $R$ is a Noetherian ring, and $x$ is a single variable, then $R[x]$ is a Noetherian ring.
The original(?) form
If $F$ is a field and $x$ is a tuple of variables, then $F[x]$ is a Noetherian ring.
This second form sounds nearer to logic / model theory.
Still, it is a slippery terrain....