Now how to shoot fast ?

[Gun Kraft]

Does split time matter? Probably not for self defense (I carry a P365 daily)

Actually split times do matter but maybe not the way you think. You don't want "jailbait split times". Think 0.19s or faster. The reason is you can't process what's happening that quickly so you may launch rounds you don't want to.
It's not just being able to articulate an imminent threat every time you pull the trigger, although that is important. I heard a story of one LE at close range intended to shoot a Mozambique at a threat. He was so fast that he missed the head shot because the suspect fell after the two body shots. He would have made the head shot if the suspect was still standing but couldn't adjust that quickly to a dynamically changing target.

[arrix]

Practice dry firing with a laser bore sight until the dot doesn't move around.

[G-forceJunkie]

Practice dry firing with a laser bore sight until the dot doesn't move around.

This can help, but also build the bad habit of not watching the front sight. You need to train your brain to pull focus back to the front site anytime your pressing the trigger. Pressing the trigger hundreds or thousands of time while not watching the front site is counterproductive. Now doing this exercise while having someone else watch the laser, or video record and review after would be much better.

[CAL.BAR]

In short.... practice. It takes a few thousand rounds to get good and accurate while shooting slowly and stationary. It will take ten thousand rounds to get really good at shooting accurately and fast. Add a few tens of thousands more and you will be good at shooting quickly, accurately, and.... while YOU are moving as well. Last I heard, the top IDPA shooters run though about 30-40 THOUSAND rounds A YEAR to stay on top.