This is how I understand it:

Please note: I am not sure if the increased boss time provided by Savior Shield is added to the formula for determining if your idle heroes can kill the boss.

While letting your heroes idle you CAN safely tap AFTER 25 seconds have passed in the battle.

The chart that you referenced is just a rpresentation of your odds of hero deathduring a boss fight. Mini Bosses (stages ending in 1-4 or 6-9) have a 5% chance to kill your heroes and Bosses (stages ending in 5 or 0) have a 20% chance to kill your heroes. The "Early Start" strategy assumes that you are trying to kill the boss in less than 5 seconds, thus If it takes you 5 seconds or less you have a 0% chance of being killed, but if you were to take 6 seconds you would be entering that 20% chance, but the 20% chance is spread across 25 seconds. So during the first second your odds of dying are 1/25 * 20, which equals about 0.8% chance of dying at 6 seconds into the fight. Therefore, the longer you press your luck, the better chance you'll have a hero die, up to a maximum of 20%. The "Late Start" strategy is just the inverse, it assumes that you are going to let your idling heroes provide damage to the boss for you and you can start tapping during the last 5 seconds to kill the boss. But let's say that 5 seconds of tapping plus 25 seconds of hero DPS just isn't enough, well you can be risky and jump in BEFORE the 5 seconds left mark. The chart tells you how much risk you'll be taking by starting before the last 5 seconds.

As for which hero the game decides to kill, well I believe that is a completely random event. I have had the game kill my strongest hero that I just finished saving up for, I've had it kill Takeda while I was already up to Jaqulin, and I've had it kill anyone in between.

I hope all of that is helpful to you and everyone else who is confused by hero death.