Summon FATE - global cooldown?

There is a head talisman (can’t think of the name right now) that summons FATE to do damage to a target you hit 3 times in succession.

There is a Luck talisman (Anima Infused Keepsake) that has a chance to summon FATE on a critical.

Both have 20 second cooldowns. The damage increases with talisman power as far as I can see but not the cooldown.

Is this the same cooldown? In other words, if I have both equipped and summon FATE by hitting 3 times, does this remove the chance of summoning FATE on a critical?

I have the Head talisman currently equipped and I just received the Luck. I’m trying to figure out whether it would be worth using it or if I should sell it instead.

They are not both FATE. Head is FATE Luck is Gatekeeper. They act independently of one another.

Oh, OK.

Once again - not reading closely enough. Sorry.