Wait....I've never heard of a different Organization...I mean Wesker & Ada were working in the same one. I have Archives(you are referring to the large book that is like an RE encyclopedia right? Has stuff from RE0, REmake, RE2, RE3 & CVX.), and I don't recall anything like that.The 3rd Organization is mentioned in Archives and some guides, and it is the only "Organization" Wesker worked for. It is different from "The Organization" Ada worked for.
Let me explain a bit better: Wesker joined the 3rd Organization right after RE1 becaused he offered them data and biological samples. He met Ada and worked with her to retrieve the G virus. However, i'm not sure if she was REALLY working for the 3rd Org or spying for The Org. Wesker continued to worked woth the 3rd Org, and in 2004, he invited Ada to work with him once more. However, this time, she was spying for The Org and had orders not to deliver the Control Plaga to Wesker (through he kinda predicted it and somehow obtained a dead control plaga).
So, there's two "Organizations": Wesker's ans Ada's. The Org seems to be benevolent, i mean, not wanting to develop BOW's, but to study them instead.
"S" is Tricell, they used that letter as a codename for "SeaShell", Tricell's original name (they changed it due to the "Shell" oil company).
http://projectumbrella.net/articles/3rd-Organization
http://projectumbrella.net/articles/Organization
S, I've heard is Tricell, though I haven't heard any confirmation anyway. I guess that stuff about SeaShell and using all that as a codename makes sense, though I don't remember it in the games, it probably wasn't in there was it?
That's what annoys me, that Capcom neglected to tie up those loose ends, with The Organization, S, Tricell, etc, etc.
Now pretty much everything you said makes perfect sense, about the organization and it's intentions and all, but I don't recall there being separate ones, that is, one that Ada was with & a totally different one that Wesker was with.
I guess the only question is, why do you think there are two separate ones? I mean what would imply that, and what source might you have? You mentioned Archives, and I'll go back and look, but IDT that is in there.