Since 3125 is just a collection of ideas, it's possible that as it corrupted humanity, human ideas also became part of its memeplex, fundamentally altering it to include a sense of self-preservation and understanding of communication. (The rewrite goes into more detail about ideatic space and how everyone and everything exists in it as concepts, grouped together in various ways.) Alternatively, Red voluntarily embracing 3125 pre-"Ojai" could have been the only thing that instilled human-like qualities into it. Hard to say.

Regardless, Red certainly seemed like a unique part of 3125, especially since he's described as "the piece of SCP-3125 which was capable of communication", and killing him removed that part of 3125. Maybe joining voluntarily was the difference, or maybe he had some sort of antimemetic-blocking property like Adam did, idk. He also seemed to retain more autonomy than other corrupted humans, but maybe 3125's concepts just filtered through him into something that sounded human (holding a grudge against Marion for shooting him at the lake, talking about "Us" and "Them", etc). Either way, I don't think Adam killing Red harmed 3125, per se, just changed the composition of the memeplex slightly.

(Does any of that make sense? My brain hurts now.)

Also, the gun Adam shoots Red with is changed in the rewrite to be SCP-7381, the ray gun that deletes matter, if that helps. Can't shake off a bullet to the head if you no longer have a mind to conceptualize away the harm (or whatever was keeping him alive after the lake).

Edit: Maybe the unique quality Red brought was the idea of using communication to spread 3125; maybe 3125 included the concept of "communication", but not "communicate 3125 to other things".

humanity wasn't learning about it, they were creating it as an entity with their exhaustive research and memory enhancing drugs. It's the old "Don't think about the pink elephant" trick gone rouge because the habitual abuse of these memory enhancing drugs will create a concept so detailed that it takes on a life independent of your own, and in this case their imaginary worst case 'what if' is literally an idea that sprang to life and ate them like a manifest collective anxiety attack with real force behind it. Hitting it with a human memory glue bomb effectively crystalized it into being an entity in the past rather than an entity still moving through our present.