He has a hidden stat, call it affection if you want. How high it is when you get him is determined by how much you payed him to join you. Everytime you pay him in battle the stat goes up or down, depending on how much you give him. This stat greatly influences the odds of zanmato. I believe if the stat is too low it's not even possible for him to do zanmato, and when the stat is max and you give him more than a certain threshhold he has like 96% chance to do it.
I'm not sure about the exact numbers, but I will look it up now. I will edit this answer in a few minutes with the right numbers.
Edit: it turns out there is a lot more involved than I remembered. It's too much to post in here, but you can find it all in Azure_edge's guide under the chapter 'attack mathematics' (2.03).
There isn't one definitive answer for which is the best Overdrive mode since it's dependent on the character and who you're fighting.
For grinding out regular enemies that you kill pretty quickly, Warrior or Slayer is probably your best choice since you'll be doing most of the damage.
For enemies that hit you with more powerful attacks, Stoic or Comrade might be your best option for your damaging characters whereas Healer could be used for Yuna.
I would avoid using Ally since it charges very slow. I would also avoid Coward, Hero, Daredevil, Tactician, Avenger, Sufferer, Victor, Victim, Solo, Dancer, or Rook since none of these are very well suited to grinding.
You can find a complete list of all the different modes here along with how to make use of each mode.
Best Answer
If you're going for SP grinding, there's a super easy way to go about it. The simple answer is to use Stoic for one, and Comrade for the other two; all the damage the character takes, even beyond their maximum, goes straight into Overdrive, and you'll get less for the others, but still a decent amount.
The slightly longer answer is to get everyone a weapon with Overdrive -> SP, and Triple Overdrive and Triple AP. Getting both will make it faster, but isn't an absolute requirement. One of your Comrade members should also have Auto Phoenix (if you're still at a fairly weak state. Optional if you're just trying to wrap up the sphere grid).
You then want to fight Master Tonberry in the arena. Your member with Stoic will attack, and the grudge will kill them. If you have Auto Phoenix on one of the others, you'll resurrect automatically. Otherwise just use a Phoenix Down. Either Comrade can cast Haste on the one that died, but should not attack. Repeat. If you can't kill the Master Tonberry before he gets too close, run. You'll still get a ton of AP, and will probably hit the max without too much issue if you do manage to kill him.