Hello Nathaniel!
Kenshi is the official term, Shorinji-ka isn't a word (that I know of).
Your guess about the monk's robes is spot on: Shorinjikempo is based on Kongo Zen Buddhism and retains* some rituals from it. Hence the use of the hoi for formal occasions like demonstrations.
I'm not sure about the different kinds of belts, I've only seen the official one (the cloth one, I suppose?), which we like to call "sausage". Personally, I'm not a big fan of it and maybe others agree and switched to using ropes instead …
*Within Japan itself, almost exclusively, Kongo Zen is in fact an actively practised religion with ceremonies like marriages etc. Outside of Japan it's next to impossible to get ahold of this side of Shorinjikempo, but the philosophy that is at its base heavily leans on Buddhism and the techniques reflect that, too. Therefore the use of the hoi on formal occasions is a good reminder of this heritage, IMHO.
Hope that helps!
Jan
Jan Lipsius
少林寺拳法
Shorinjikempo
Humboldt University Berlin Branch
"An eye for an eye only ends up making the whole world blind." Gandhi