In the default config, when you suicide, it's like the player killed themselves as a player and says it in the chat.

How can Suicides not have anything written in the chat?

Please with sugar... ;)

You need to define a message block with "Suicide" defined as damage type.

Such already exists in the default configuration unless you removed it:

{
	"KillerType": "Player",
	"VictimType": "Player",
	"DamageType": "Suicide",
	"Messages": [ "{victim} had enough of life." ]
},

There simply clear the "Messages" collection of any messages:

{
	"KillerType": "Player",
	"VictimType": "Player",
	"DamageType": "Suicide",
	"Messages": [ ]
},​