Personally, I never understood why it matters, outside of you own school and organization. Who really cares what some other school or organization thinks of my rank?
I watched both of your videos that you posted here. I can tell one thing right away. None of the people in the videos have trained Danzan Ryu Jujitsu. If any of them came to my school, they would start as white belts. I'll give a concrete example of why. The falls demonstrated are not Danzan Ryu falls. This matters, because Danzan Ryu throws require uke to take the correct fall, so that uke does not get hurt. There are details to our falls that match up with details in our throws. Now the throw will work just fine... I would just need to find another uke.
Now, let me turn this around. If I posted video of myself doing Danzan Ryu, you would learn very quickly, that I have no training in your art. (I think the sign on the window in your video said Karate) If I came to your school, what good would it do me to strap on my black belt and demand that every start "Siring" me? Not only can I probably not do your white belt katas... I don't even know what they are. If I wore a black belt, but had no clue about your art, might I get hurt trying to participate with the black belts?
In my gi bag, I carry my black belt and a white belt. In my school, I wear the black belt. If I go to an AJJF event, I will also wear my black belt. If I go to another Danzan Ryu organizations event, I hold both white and black in my hand, let them know my rank in the AJJF and ask which they prefer I wear. When I go to any other art, put the white belt on. I let them know I have been training a number of years in Danzan Ryu, but want to be treated like any other white belt.
My belt, my rank, really only mean things in my school and my organization. So, my school records and my organization records are all that should matter. If I am teaching Danzan Ryu, who cares what the kukkiwon thinks of my rank? If I were to teach TDK who cares what the AJJF says about my rank?