"Everyone permanently resident in the European Union is covered under the national healthcare systems."
not correct! not everybody is automatically insured! one must have payed a certain time into the health care system in order to have permission for it!
so it can be covered either by working or as a pensioner(both automatically insured) or unemployment benefit entitlement or other social eligibility, otherwise you just have to pay yourself a monthly deposit into the health system in order to obtain power.
If you have a card of an EU country so you can use by travelling in other EU countries. sometimes can be that in some countries is to pay directly at a medicine doctor or hospital and you can retrieve then later in the country where the card was issued.
to see at the backside of health-care-card:
Foreign health insurance certificate replacement
The back of the e-card is designed as a "European Health Insurance Card" (EHIC) and within the EU Member States, EEA countries and Switzerland valid. This eliminates the issue of the old "holiday sickness certificates".
http://de.wikipedia.org/wiki/Verordnung ... r._1408/71
When you live permanently then in another EU-country you must look there for the health-care-card and you have same procedure. It's always after primary residence.
And the systems are often different in different countries - some ways to offices until to get the health card in hands.