Order isn't going to matter a whole lot, really. I'm sure the factory service manual does have a preferred order so if anyone has that, no reason not to follow it.
I have gotten better performance over a long time period from readily available off-the-shelf name-brand (Castrol, Pennzoil, etc.) DOT4 or DOT5.1 from parts stores than some "race" brake fluids, including ATE type 200. I had to rebleed ATE200 after one track day in the s2000, vs. as many as 6-8 track days with off the shelf stuff before doing a caliper bleed. Most likely you don't need or want anything fancy here.
FWIW I do the one-man bleed method using this:
Loosen bleed screw, pump pump pump, top off reservoir, repeat for all 4 corners.
Idea the first time is to completely flush out the old fluid, so you have to pump until reservoir gets somewhat close to empty, top it off and pump pump pump again until you get fresh fluid in the calipers. After that you can likely just caliper bleed the next time the pedal feels soft, and flush again beginning of next track season.