The designee is the person you are authorizing to care for your child. The parents will designate who they want to care for their child while on board. In your example, you are the designee.
Health insurance is not required by Disney. In the US, an insurance card would make it easier to get health care in port if necessary. I'm not sure how things work in Europe (or in the Caribbean, for that matter).
On the ship, passengers are billed directly for any medical charges and can then submit to their insurance company (if they have one) for reimbursement. Again, I'm not sure how reimbursement would work for people from areas with different health care systems.
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