OK. The problem is solved! I took the car to AutoZone for a test. It passed all the tests for battery/alternator/starter. All in, I spent $225 for the alternator and $18 to get a new battery. My battery died one month short of the 3 year warranty. Since when I bought the battery it was 3-year full replacement warranty, I got a new one nearly free. (But if we buy from Costco now, it is pro-rated 3-year warranty.) So considering I got a pretty good deal on the battery, it doesn't seem to be such a bad occurrence. Also my alternator died in a pretty dramatic fashion relative to others. So the diagnostic part is actually pretty much non-existant.
There is such a great satisfaction to get things done by oneself. Otherwise, justing towing the car to the dealer or a shop would be almost $100. (San Mateo is a pretty expensive place in the whole coutnry.) And I don't even know how much it is nowadays to get it fixed with the requirement to use a Denso alternator. Considering just the alternator without coupon is more than $270, I guess the bill wont' be lower than $800, right?
For the actual work, I followed this video exactly. This kind of almost professional production is indeed much better than normal videos made by mechanics or home warriors. After I dug out the old battery. I drove it back home and get the rest done in my garage. It would have been a gigantic hassle if I have to reinstall the serpentine belt in the parking lot. The drive home was pretty scary because power steering was gone. Expect to turn the steering wheel like a mad man. Thankfully, I have only 0.5 miles to cover.
The only thing I want to point out to the fellow MDXers in the repair process is that Denso replaced the stud on the top nut with a bigger one. The one I got from the old unit has a 12mm nut. The new stud requires a 13mm one. It took me a VERY long time trying to put the 12mm nut on the new stud without luck and eventually I lost the nut in the engine bay. Then I grabbed a spare 13mm nut to put it on and it worked in the first attempt. In this aspect, Denso is actually not as nice as other reman/aftermarket unit like the one in the video. They came with a new nut. I was not experienced enough to try it out before the alternator was put into the car. Lesson learned.
It is really sad that I always planned to do a real thorough alternator/starter DIY and take a ton of pictures and put them on the forum to share. But when it really happened, it was not an option. Hopefully, I can do a better job next time. Happy motoring guys! Thanks for the help.