Translating HORNSAT to SAT, revisited See Translating SAT to HornSAT 

It says in that post that it is impossible to translate SAT into HORNSAT. But it is known that HORNSAT is P-complete, so any language in P can be translated into HORNSAT. Then if it is impossible to translate SAT, which is in NP, into HORNSAT, then wouldn't this imply that P is not NP? What am I missing?
That question was about a translation from an arbitrary instance of SAT to a HORN-SAT instance with the exact same satisfying assignments. The translation property needed for P$\neq$NP is just equisatisfiability, not equality of assignments. That is, it simply requires them both to be satisfiable, or unsatisfiable. If satisfiable, they may be satisfied by different assignments.

For example: 
$a \wedge \neg b$ and $\neg a \wedge b$ are equisatisfiable (they are both satisfiable), but they have different satisfying assignments.

The answer was that there can not be a satisfying assignment preserving translation, but this doesn't tell us whether there is a polynomial time satisfiability preserving translation.