I am going to try and offer some info that I hope will be useful to you. Although the Savage is a simple beast, it’s electrical system seems overly complicated to me. I will try to give you a better idea of how the current flows.
The first piece of advice I offer is about nomenclature. Your vid was nice, it helps us see your problem. Correct nomenclature is very important. I suggest when you want to verbally identify a part, go to one of the on-line parts distributors and find the illustrated parts breakdown for your LS650. Find the part in question and use the nomenclature they provide. The little black box you replaced is not a fuse, it is a “Decompression Control Unit”. Here’s a link to Babbitts Suzuki Parts House. It’s a good tool.
https://www.suzukipartshouse.com/oemparts/c/suzuki_motorcycle/partsThe second piece of advice I want to share is that a battery that only has 11.7 volts of potential is “DEAD”. I concur with JOG & Versy. A fully charged battery must have 12.7 volts of potential. That being said, the fact that you can jump directly to your starter and get the starter to crank the engine tells me that you should at least get a little action when you hit the starter button. So, aside you’re your tired battery, something else isn’t right.
The electrical power for all of your components runs through two fuses. Those fuses are identified on the wiring diagram as “Fuse 1” and “Fuse 2”. They are both 20 amp fuses. They are located under your seat in a small white plastic box. I can see the box in your video. This is the fuse box in question.