You still need to land on SSR Hime slot to get SSR Hime (which we assume to be equally low on both servers), but Rate Up on DMM is on a completely different level to Nutaku.
Honestly, it'd be nice to see Nutaku "Rate Up" modifier. I wouldn't be surprised if it was 2x normally. The current gacha is all happy going OMG FIRE HIME UP TO 3X HIGHER!!!" while DMM started at 5x and at this point was at 20x. (Currently at 25x I believe)