Computational Complexity of 'Generic'/'Relaxed' Horn 3SAT
Schaefer's dichotomy theorem gives you the answer. Apply it and see what you get, and let us know.

Looking at the modern formulation, in your case $\Gamma$ contains the three relations $\lnot x \lor \lnot y \lor \lnot z, x \lor \lnot y \lor \lnot z, x \lor y \lor \lnot z$. For each of these relations $R$, you have to go over the list of polymorphisms, and check whether one of them is a polymorphism of $R$. For example, to check whether the binary AND function is a polymorphism of $\lnot x \lor \lnot y \lor \lnot z$, you have to check whether
$$
(\lnot x_1 \lor \lnot y_1 \lor \lnot z_1) \land (\lnot x_2 \lor \lnot y_2 \lor \lnot z_2) \\ \Longrightarrow \lnot (x_1 \land x_2) \lor \lnot (y_1 \land y_2) \lor \lnot (z_1 \land z_2).
$$
If every $R \in \Gamma$ has one of the six polymorphisms, then your problem is in P. Otherwise, it's NP-complete.