The cooldown duration is tied to the weapon that executed the proc, so the order counts.
If Weapon A procs, it's implicit that Weapon A won't proc again 'til Weapon A's proc has expired.
If Weapon A procs and you switch to Weapon B, Weapon B won't proc until the proc period of Weapon A has expired.
If after Weapon A's proc expires, you proc Weapon B and then switch back to Weapon A - Weapon A's proc won't happen 'til Weapon B's cooldown has expired.
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