Yes, I think that's a good summary of secondary actions.
The part of the mountain the oni is on isn't as steep and doesn't require an ST if running. (The PC's didn't need STs running there either.) From the area below the earthquake to the bottom of the map requires a Speed ST if running straight down the slope directly away from the oni.. If traversing down the slope at an angle, no ST needed.
Turn 3
Phase 13 - Jinsei in effect Walks 8 yds down the step slope, no Speed ST needed.
Phase 10 - Fuma stands (Alter Position primary action).
Phase 10 - Minoru Holds his primary action to any phase down to 5. He'll go before Kaida's primary action in 4 or take no action. (RP-wise, that means that Minoru will never know what Kaida did before taking or losing his action. I don't know if we want to discuss this rule or just leave it as is).
Phase 8 - Jinsei secondary - None, unless he wants to ready a weapon, etc. His movement happened in phase 13.
Phase 5 -
Oni runs 6 yds down the slope.
Fuma shoots han-kyu shot at oni, misses.
Minoru secondary
Jinsei secondary - None, unless he wants to ready a weapon, etc.
Phase 4 - Kaida casts Pure Flesh on self, heals 4 Health.
Phase 2 - Kaida parries.
Phase 1 - Effects phase. No effects this turn.
Range to oni
Jinsei - 24 yds.
Minoru, Kaida - 15 yds.
Fuma - 12 yds, but behind Kaida's mud pit.
Minoru can do his action, or if he spends it observing, we can go to Turn 4.