It would take way too long to describe how the proc works accurately with pictures because of the complicated nature of timelines, but I'll try to do my best.
First, just to make things clear: The Crescent's Shadow only has two attributes beyond its base damage: Crescent Blade (a proc), and a 100% crit. dmg bonus modifier. It also has a slot for a grip and a rune. Nothing more.
Now, here's how the proc works.
The timeline found in DA3/Party/General/Passive/unique_item_passive_proc_crescent_timeline creates a rule to call the timeline in da3/party/general/passive/unique_item_passive_proc_crescent_check_timeline every time the character deals damage (ReactionType: React_OnDealtDamage). In other words, it will try to proc every time you deal damage of any kind.
Now, everytime you deal damage and the DA3/Party/General/Passive/unique_item_passive_proc_crescent_check_timeline timeline is called, this timeline will trigger another very short timeline (0,033s) that will attempt to call another timeline in da3/party/general/passive/unique_item_passive_proc_crescent_triggered_timeline, but with a conditional check: it will only call the "triggered" timeline if the RNG (with a random seed) falls within a 20% chance. Meaning it only has 20% chance on hit to proc.
If the random chance test succeeds, the timeline in da3/party/general/passive/unique_item_passive_proc_crescent_triggered_timeline is called, the game will call upon two [BWCSMAction_LaunchNewProjectile] actions (in other words, two cold bolts) and there's a lot of stuff about projectile math that isn't worth discussing about (initial velocity, ground clearance, max bounces, etc.). These projectiles are allowed to be launched with a late start. Then the timeline finally calls the da3/party/general/passive/unique_item_passive_proc_crescent_cooldown_timeline (along with another timeline not worth discussing). This "cooldown" timeline has a fixed duration of 8.000s (ConstValue : 8,000)
In other words.
On any damage you deal, you have a 20% chance to proc 2x cold bolts. Once this happens, a 8s internal cooldown immediately triggers, even if it may take 1-2s for the cold bolts to hit the enemies because of late start.