If you booked directly with HAL, a Cruise Confirmation is emailed to you. After final payment and if you purchased their Cancellation Plan, the final Booking Confirmation will show under Guest Information that you have the Platinum Cancellation Plan.
I have had to cancel a HAL cruise several times prior to sailing, and upon calling the HAL reservation number, the agent knows I have the plan and arranges the refund which I have normally received within the week.
For cruises that I have sailed on, and had to cancel mid-cruise due to an emergency at home, once HAL is notified, the pro-rated refund comes through after notification.
As for health related emergencies, my only experience is that I have had to pay the bills and then file for reimbursement after the fact.
HAL does know who has their Platinum or Standard Cancellation Plans.
If you have booked with a travel agent, I don't know how it works. I suspect you have to go through your agent. Others here know how that works.