Videos: Headlight brightens and dimms-charging system test

"there was default ‘0L’ on multimeter display. Are you meaning to this?"
Yes OL is a high/infinite resistance, which is what it should be.

I think your R/R should be 32800-47H00. I'm used to the black epoxy on the underside but after looking at several on ebay (lots of mislabeled parts there), I now believe that yours should look like this. The eighth image shows the underside as a metal plate. This is new to me. But you can see that it's a Shimaden SH811AA.

Per the service manual, the voltage regulator test is the regulated voltage inspection that you performed above, and which failed.

I personally don't have a problem with your battery but opinions vary. The only thing that I'd avoid is a cheap battery from China.

P.S. Search for "What are the symptoms of a failing voltage regulator?" Note too high a voltage and dimming/pulsing lights.

You can buy a new 32800-47H00 for about $200 and be done with it. You can buy a used 32800-47H00 on ebay for $50-$60 but you have to be careful there as many are mislabeled. The real ones are stamped 47H00 and SH811AA and have the stainless plate on the underside. Also, unlike the earlier Kokusan Denki R/Rs, the heatsink fins don't have circular risers or ejector lands (or whatever they're called) on them.

MOSFET I think of them as low resistance transistorized switches but may be somewhat off base there.

You can buy a Shindengen FH020AA here for $100+. The real ones are stamped with the Shindengen logo and FH020AA and the stainless plate isn't held on by screws. The cheap ones there are highly suspect. Ironically the $26 AliExpress R/R that I link to above has the right markings & plate. You could try asking him if it's a real Shindengen.

The FH020AA has 70 mm mounting centers vs 80 mm on the Suzuki so you'd have to deal with that. It might be as simple as elongating the holes in the R/R or maybe those on the bracket that supports it. You'll also have to make adapter connectors. The GHC R/R linked above supposedly has extended mounting holes. They're in Taiwan instead of mainland China and seem more believable. But I can't find any specs or distributors.

P.S. I don't claim to be knowledgeable about this but think that Yamaha started using the Shindengen FH012AA around 2006. It was later replaced by the FH020AA. I don't know why Suzuki hangs onto the older technology as, per the Wikipedia article, MOSFET technology is definitely superior.

P.P.S. The ebay seller of the FH020AA says the plate is stainless. But here's a Shindengen data sheet for the FH012AA that says it's aluminum. I don't know why the later part should be different.