I think there's plenty of examples in the game that show how the Turks aren't evil monsters and are just everyday people like you and me, albeit with less qualms about having to commit shady deeds then most people would have. To give a few:
There's the scene in Gongaga where Reno and Rude (but mostly Reno) are gossiping about relationships and who likes whom, showing that yes, even the guys who do Shinra's dirty work engage in friendly banter and silly gossip with each other.
There's the times you encounter them off duty getting some R&R in bars and such, where they(except Elena at first, but Reno talks some sense into her) refuse to pursue your party because they're off duty, showing that it's nothing personal for them and merely a job.
And of course there's the Wutai sequence, where they help they party rescure Elena and Yuffie from Corneo, and afterwards decide they're off duty and let the party go about their business. Reno and Rude even say they went through the trouble of teaming up with Cloud and co. to get Corneo because it's they're job.
So yeah, the original game makes it abundantly clear that the Turks are just punch clock villains, and not actually evil people. That's why they're on better terms with the party after the OG, since it's no longer their job to oppose the party, they have no reason to bear any feelings of adversity towards them.(not that they[especially Reno and Rude] ever really did in the first place, except maybe Elena after Tseng was wounded, which even then just shows that she's a human being who cares about her comrades)