It's hard to tell who's faster now with the purify system, used to be a lot easier you can just look at their refine lvl. With purify you can easily add 1k+ to initiatives. The person you were fighting probably has higher initiatives than you from purify.
As lightning I stack initiatives, it's the only stats that matters, so I just keep reroll til I get high initiatives. My initiatives is close to 3k now.

I see what you mean but I can assure you that my purify is greater as well. So it comes down to: Does positioning play a greater roll than initiative or does auto play a greater roll? And either I'm missing something or I should be faster.

Move numbers > Initiative > Prompt skills > Non-prompt skill

Regardless of refine or purify, position plays the most important role. If you were spam clicking your ninja in position 3, no matter what, the opponents tobi in 2 is going to skill faster than you. Initiative only comes into play if ninjas are competing from the same position.