I'm answering my own question since in the meantime I experienced the whole procedure and I can speak about what actually happened.
Unfortunately, the DIBP website is confused regarding the health cover requirements for the 408 visa. Despite the message that you need a private insurance "if you are not covered by Medicare", in fact Medicare coverage is often not enough.
Let's start from the official web page for the 408 visa, which says (emphasis mine):
You must make adequate arrangements in Australia for health insurance, for the period of your intended stay in Australia, unless you are covered by Medicare. You should ensure that your insurance will provide at least the level of coverage required for the purpose of your stay. Your health insurer could be in either Australia or your home country.
Attachment A in the health insurance standard template letter (142KB RTF) is a guide to the level of health insurance we will accept as adequate.
So there is a minimum standard of services that your health insurance must cover, and you must prove that your insurance covers what is required, not only that you are or are not insured. The linked document is a template of a letter that your insurance company has to sign to state that your insurance covers what is needed. This is the document that you need to upload during the visa application web page to have your health insurance requirements cleared.
Now let's see which health services are listed in the document. One of those is Ambulance services:
Ambulance services – 100% of the charge, that is not otherwise covered
by third party arrangements, for transport by ambulance provided by,
or under an arrangement with, a government approved ambulance service
when medically necessary for admission to hospital, emergency
treatment on-site, or inter-hospital transfer for emergency treatment.
However, ambulance transport to hospitals is not covered by Medicare under international reciprocal agreements, as stated on the Medicare website (emphasis mine):
It doesn’t cover:
- medicines that aren’t in the Pharmaceutical Benefits Scheme
- Pharmaceutical Benefits Scheme medicines for In Vitro Fertilisation
- (IVF) care planned before you got to Australia
- care for Assisted Reproductive Technology (ART)
- private hospital care
- public hospital care as a private patient
- ambulance travel
Other things may be missing besides ambulance travel, but I didn't check every item in the list. The net effect of this mismatch is that the Medicare reciprocal agreement with most European countries is not enough for the 408 visa, and you'll need a private insurance policy. Honestly, given that an ambulance transport can cost even thousands of dollars in some Australian hospitals, I'd have done it anyway.
I've not inferred this fact alone, but it has been confirmed to me by phone calling the London DIBP call center, and also now that I'm leaving the country, my private insurance company required me to show my departing flight ticket before they allowed me to cancel the subscription.
So people applying for a 408 visa have to enroll into a private health insurance policy. The good news is that this is so common that most Australian insurance companies already provide the right kind of policy for the corresponding visas. I don't want to advertise any company in particular, there are plenty of them, but the important thing is to do it with an Australian company and not one from your home country, since the prices can be much different, especially if coming from Europe (Australian health insurance market is much more competitive so prices are lower). Also all the procedures for claims and reimbursements of expenses are much easier with Australian companies since they're of course integrated with the system.
The keyword to look for is OVHC, which stands for Overseas Visitor Health Cover. Looking for offers from Australian insurance companies under this keyword you will find what you need.
Most companies offer an online procedure to subscribe which let you specify your visa type (e.g. 408 in this case) so they can configure the policy to automatically meet your requirements. After the online subscription procedure, they will directly email to you the health cover letter (a well-written version of the template linked above) that you can directly attach on the visa application web page.
Prices are fair enough, varying from A$75 to A$150 per month depending on the company and the additional services (e.g. whether visits from general practictioners are included), so it's not really a big deal.
Besides what is required by the visa, I'd recommend such a policy to anyone because, especially coming from Europe, the Australian health system can always present some surprise. For example, a package of contact lenses for one month, which would cost 20€ in Italy, cost as much as A$120 in Australia, plus the cost of the mandatory visit to an optician, since here they can only be sold on prescription. A private insurance would happily cover that (but not the most basic ones, so check your conditions). Similarly, a lot of drugs that are sold freely in Europe are sold only under prescription here, and visits to general practitioners are only covered partially by the Medicare agreement, so depending on one's general health situation, spending ahead some A$100 per month can save quite a few headaches.