Hey everyone, thank you all for following up! And PainlessCandy, thanks for the video; I've been looking for an animation of the helical LSD for a while. I found an Integra forum where somebody had a picture of the helical LSD, which was very similar in principle to the video you posted. Main difference I think is that the one in the video is for RWD, and ours is FWD differential.
First of all, the summary: I changed the MTF and the noise completely went away.
Long story:
I took it back to the dealer and the service manager heard and recorded the noise but told me that the noise needs to get much louder before he can diagnose it up on the lift and told me to keep driving it. I asked him if changing the MTF would help and he said that they never work on manual cars anyways and the MTF shouldn't need changing. He also didn't seem knowledgeable about Si's and LSD's.
Then, I ran into Scotty Kilmer's "Howling Honda" video on Youtube where he says that the CRV rear LSD's are notorious for a howl during sharp turns and the fix is to change the differential fluid. After seeing that video, I did more research and came across Honda Element forums where people said that after changing the MTF, all kinds of weird noises went away. I also saw more people talk about CRV noise getting fixed after the differential fluid replacement.
So, even though the Honda MTF and the Dual Pump II differential fluid are different stuff, they both are gear oils for Honda LSD's. I was determined to replace the MTF on my car.
What drained out was a very dirty MTF. Even though the fresh Honda MTF has a dark oily red hue to it, what came out was practically black, even in a white, Dollar Tree drain pan. Luckily though, there were no chunks of any kind. It's been 4 days since I replaced the MTF and there is no noise. Now, I have no idea why the MTF was so dirty, but I used to use engine brake excessively, so I stopped doing that. I don't otherwise abuse my car.
Changing the MTF was as easy as changing the oil. I cut the flap of a large cardboard box, folded into a vee, wedged it between the transmission and the frame, and let it sit in the drain pan. When I removed the drain plug, the gear oil just hit the cardboard and went straight into the pan, without making a mess. I then refilled the transmission from the top fill hole, under the air filter box. The filter box comes off pretty easily after pulling on it hard (harder than you would think).
I would highly recommend changing the MTF relatively regularly. I plan to do it again so to flush out the contaminants and then do it every 15k or so.