You can’t explain it to me because you’re wrong — and you’re wrong because you do not understand the canonical (correct) definition of spam and why it deliberately omits certain qualifiers.

For example, the canonical definition of spam (UBE) does not discuss who is sending the email, why they’re sending the email, what the content might be (or even whether there is any content at all). These omissions are not an accident: they were done with purpose decades ago when the term was coined, and when we were all engaged in numerous (and lengthy) debates about how it should replace earlier extant terminology (e.g., “mass mail abuse”, which was in use during the early 1980’s).

It also does not include word “commercial” — another deliberate omission. It is a newbie-level mistake to presume that non-commercial UBE can’t be spam…because of course it is. (Do not conflate the canonical definition of spam with the incorrect ones written into statutes in various jurisdictions. As far as I’m aware, no legislation anywhere in the world has actually used the correct definition of spam. This is unsurprising: not only do legislators wish to exempt themselves, as they’ve done from telemarketing statutes, but the legislative process is heavily influenced by pro-spam lobbyists. Here in the US, the DMA is largely responsible for the CAN-SPAM bill which essentially legalizes spam.)

Non-commercial UBE from political, charity, non-profit, religious or other entities is still UBE therefore still spam — by definition. (And of course all of us who are familiar with the long and sordid history of spam and telemarketing and junk faxing are painfully well aware that spammers will go to great lengths to claim otherwise.) This is, incidentally, why it is a fundamental error to use the term “UCE”; while the email in question may in fact truly be commercial in nature, this is completely irrelevant in determining whether it’s spam or not.

Your insurance company analogy does not hold: the customers have solicited that email traffic by agreeing to subscribe to the “late payment notification system” or whatever the functional equivalent to that is. So while it’s bulk, it’s not unsolicited, therefore not spam.

Your government (traffic ticket) analogy does not hold either, but for a different reason: it’s clearly spam *unless* the drivers in question have signed up for a “traffic ticket notification via email system” or whatever we might choose to call it. The government does not get a free pass to spam merely because it’s convenient or because it has something it wishes to communicate or because it has notices to issue.

Now…this particular case, the one we’re talking about involving Hadopi, is very clear: the messages are obviously UBE. We need not concern ourselves with who is sending them; why they’re sending them; what’s in them; or any of the rest of the irrelevant details. Unless Hadopi is running some kind of “sign up here to receive notifications if someone is accusing you of being naughty” service, and I see no indication anywhere that they are, they’re spamming.