I get what both of you are saying, but the fact is that Teresi was still a snitch and Tommy was letting "daddy shyt" cloud his judgment. On this show, they treat snitching like it's the worst thing you could possibly do, on the level of being a Nazi or a pedophile. Tommy was skeptical about Teresi from the beginning, but a few episodes later, everything's cool with him and he trusts him more than he trusts Ghost, who keeps telling him to be careful around Teresi. He doesn't even see him being a snitch as a possibility until he gets the pictures, and even after that, Vincent has to give him the order to kill his father. It's not even something he immediately decides to do. And then it turns out that Tommy only cared about Teresi being a snitch because he thought he was snitching on him. Which means he would have kept a relationship with his father while Ghost rotted away in a prison cell.
My point is, Tommy was so emotional about a guy he barely knew his entire life and needed to make sure him being a snitch checked out. Meanwhile, he's known Proctor for longer and decides to take care of it the minute he hears about him being the witness. Wouldn't that be something he looks into before doing anything? Especially after what happened with Teresi? No, he just needs to hear Saxe say something about Proctor and he's ready to put him to rest like some maniacal supervillain.
And yeah, Proctor was a snitch, but I'm looking at both cases from Tommy's perspective.