I guess now's as good of a time to ask, I only know "Weeaboo" from Perry Bible Fellowship and I've never understood its connection to anime or eastern culture in general. "Wapanese" was a bit offensive but no more than "Engrish".. I dunno. I'm bored.
As I understand it, it originates from Something Awful's forums. They had a word-replacement system for cussing and other stuff (if you typed fuck it would replace it with gently caress, etc.) and as a joke just to piss off anime people's they made the auto-replace turn "otaku" or something like that into "Weeaboo" which was just the nonsense word from Perry Bible Fellowship.
It originated on 4chan's /b/ board a long time ago when they still had wordfilters. PBF comics were getting posted there and the one containing the word in question got posted and the mods there decided it would be funny to filter "wapanese" to "weeaboo" and then after the wordfilter went away it still stuck and ended up spreading all over the internet.
It is both. "Weeaboo" in it's current meaning originated as a 4chan filter replacement for wap, but legend has it moot stole the word from the Perry Bible Fellowship. Also, moot was a goon and probably got his filter idea from SA, so the SA connection is reasonable.