First, there are two kinds of registration here. One is the registration of your residence with the municipality, which some states do not have. For example, there is no such registration in the UK. This registration is the reason for saying "some EU countries require you to report your presence here to the relevant authorities within a reasonable period after arrival: often at the town hall or local police station." This is analogous to the Dutch BRP registration, and indeed if you follow the link you'll see that the Netherlands requires BRP registration for EU citizens staying longer than four months.
The other kind of registration is registration with the national immigration authorities. The countries I'm familiar with don't require this at all of EU citizens except as a prerequisite for registering non-EU family members. The second kind of registration is the one that requires you to be a worker, self-employed, a student, or self sufficient. (That list comes from Article 7 of Directive 2004/38/EC.)
So, for that second kind of registration, it appears that you cannot register until you actually find a job. The right for jobseekers to remain in another EU country for longer than three months, insofar as it is present in the directive, seems to stem from Article 14 rather than Article 7, because the Article 7 list does not include jobseekers. The Article 7 categories have a direct right to remain longer than three months. By contrast, Article 14 says (with irrelevant parts edited out):
Retention of the right of residence
Union citizens and their family members shall have the right of residence provided for in Article 6 [that is, virtually unconditional for up to three months], as long as they do not become an unreasonable burden on the social assistance system of the host Member State.
Union citizens and their family members shall have the right of residence provided for in Articles 7 ... as long as they meet the conditions set out therein [that is, they're in a category from the list that doesn't include jobseekers].
By way of derogation from paragraphs 1 and 2 and without prejudice to the provisions of Chapter VI, an expulsion measure may in no case be adopted against Union citizens or their family members if:
(a) ..., or
(b) the Union citizens entered the territory of the host Member State in order to seek employment. In this case, the Union citizens and their family members may not be expelled for as long as the Union citizens can provide evidence that they are continuing to seek employment and that they have a genuine chance of being engaged.
I do not know the origin of the six-month limit mentioned in the page you linked to, but it's possible that it comes from some other law. As you can see above, the directive itself provides for a jobseeker (who can show that the search is reasonable) to be able to remain in the host country indefinitely, without a formal right to do so, as long as the search remains reasonable, by prohibiting the issue of an order to expel the jobseeker.
The upshot of this is that you may be able to register your assumption of residence with the municipality even before you are qualified under Article 7. This appears to be called cambio di residenza ("change of residence"). For example, I found relevant pages on the websites of Piacenza and Rome.
By contrast, the pages that deal with Article 7 on those sites are Attestazione di regolarità del soggiorno per cittadini comunitari ("Attestation of regularity of residence for community citizens") from Piacenza and ATTESTATO DI SOGGIORNO per i cittadini comunitari già residenti nel Comune di Roma ("ATTESTATION OF RESIDENCE for community citizens already resident in the Municipality of Rome") from Rome.
The cambio di residenza registration, it seems, ought to satisfy the organization in the Netherlands. If not, you could try asking yourself what they would accept if you'd moved to a country that has no municipal residential registration whatsoever, such as the UK, and send that to them instead.