Co Co Curry in Japan is my favorite of the second kind. It's easy to come close at home, as dole says, but the restaurant lets you pick the exact spice level, and they always get the rice just right: sticky enough to carve out with a spoon, but still porous enough that it absorbs the curry sauce.
Go Go Curry is also worth a visit, and their US franchises are reasonably authentic. But it's a slightly different style: darker sauce that's more tangy than umami.
Go Go Curry is Kanazawa-style (thick almost like gravy, served over the rice with cabbage, on a metal bowl). It's very popular and lots of places do that style. If you like it I also recommend Champion Curry and Joto Curry.
I agree the rice is quite important. I've tried making it at home with basmati rice and it's not the same.
Co Co is great in how much you can customize it. I always go for very spicy curry with potatoes and fried chicken on top.
Go Go is funny in that they list the calories of each meal in terms of rice balls. And some are like 10 rice balls! They used to have a world champion curry, limited to a few orders per day per store [1]. 2.5kg!
Coco yichiban as a few stores in the LA area that are pretty on par with Japanese locations. I visited a branch in Beijing often when I lived in China (after eating there on a trip to Tokyo).
Go Go Curry is also worth a visit, and their US franchises are reasonably authentic. But it's a slightly different style: darker sauce that's more tangy than umami.