I'm no Samurai expert, but I can analyze stuff. So just poking around on some logs, I think the problem is the same one that plagues most people:

You are simply not pressing buttons fast enough. The most recent log I could find was a Tsukiyomi kill, and comparing you to one of the top Samurai's logs...

Your fight was 11:23 and you had a total of 283 casts. His fight was 10:54 and he had 434. This means the other guy was using actions almost 50% more frequently than you -- way beyond any Skillspeed differences.

This means the problem is likely one of two things: either a lack of confidence in your rotation, or a lack of uptime on the boss. You had 187 autoattacks on the boss, the other guy had 222 -- this implies that melee uptime might very well be an issue, but it wouldn't cover all of it. Also, Higanbana, Jinpu and Shifu's uptimes are all kind of low (Shifu in particular has a much lower uptime for you -- about 60% to his 86%.)

And yes you died once but you were only down for ~3% of the fight, so that's not going to cover the huge difference in actions used.

My suggestion would be to practice the crap out of your rotation on a dummy until you can literally do it blind. In fact, I would recommend putting a book or piece of paper or something over the part of your UI thathas your actionbars while practicing so you're not tempted to stare at them. This will allow you to keep hitting those buttons while mostly looking at mechanics when you're in a real fight, as it looks like you might be hesitating on your rotation because dangerous mechanics are going off around you.