I only vaguely remember, but wasn't everyone he killed supposed to be connected to his wife's death somehow.

Edit: it was actually his kid I was thinking of. I marathon watched them all years ago and never felt the need to watch again

The TVTropes article for this seems to be Even Evil Has Standards. The idea being to emphasize that something or someone is so evil that even an evil person finds it appalling.

If OOP's characterization is accurate, you could argue for anti-hero. Seems too much for Evil Has Standards.

The Code of Harry for Dexter is this type so is say that fits for the concept of not picking random people, but I think OP is more about the drift.
That would be Flanderization, in my opinion. When one defining trait of a character progressively gets pushed to eleven until it becomes their only trait.

Without TVTropes paywall Even Evil Has Standards

it's a subset of villain decay
You have a horror antagonist, it becomes the most popular/recognisable part of your franchise, you can't make it too hateable, so it becomes less evil over time.

See also: Terminator 2

Wait, I've never encountered a paywall on TVTropes. I don't see it when I click OP's link either. Is this something new?

You have to disable your ad blocker or pay 5 bucks a month

Paywall?

That link loaded fine for me? With and without an adblocker

Same. It hasn't made a peep about using an adblocker for me, neither on my computer nor on my phone.

I wonder if it's something regional?

I did some digging and about a year ago they introduced a "please turn off afblockers :(" popup that would sometimes even show even without an adblocker. But nothing recent

Terminator 2 was a dramatic subversion though, no decay there. They told you, straight up, in-character, "we reprogrammed this evil robot to fight the other evil robot" and the rest of the movie is an exploration of morality.
"What are you doing!? I told you not to kill anyone!"
"They'll live"

I always get this, from phone and desktop

Man, that sucks. I'll bookmark the alternative you provided, since sites seem to be springing this crap out of nowhere a lot these days.

Weird, I couldn't get that to show up at all

The whole point was that the T-800 was a bottom barrel cyborg. They were all modeled after Arnold, but there were thousands of them.

They basically found a junker in the trash and reprogrammed it. He was NOT the same T-800 from The Terminator. He was a completely different robot.

The DC cinematic universe lol

Less of a villain decay but also perspective - Arnold was the most iconic part of Terminator and so the writers found a logical way to retcon the actor to continue playing the role.

In universe, John is exposed to the T800 as a hero but his mom only knows him as a killing machine. These characters have to grapple with the role reversal in story. In Saw no one (that I recall) ever said "Wait, isn't this guy just an asshole?"