I don't advise hooking it up as you have it depicted in your diagram, although it may be feasible, it's not safe. Maybe you just drew it up that way for clarity or just showing your connections, IDK, but please don't leave AC power "exposed" like that if that was your plan. Use some type of enclosure for the wall plug, and connect the relay to the plug, not directly to your 12V AC/DC adapter. After you make some kind of enclosure for your AC wall outlet and have the relay connected to it, plug the AC adapter into it (don't make any connections to the AC side of the adapter). Found this pic on google (it's pretty clear on how to hook things up), it's not exactly the same as what you depicted as it
1) uses a US outlet,
2) uses AC Hot (live - the black wire) to connect to the relays then to the bronze screw on the outlet
3) uses the neutral (white wire) to connect directly to the silver screw on the outlet
4) uses a Raspberry PI to control it
but regardless, it's essentially the same as what you want to do (just swap out the Raspberry PI for your Arduino, and the US outlet for your Euro one):

Some thoughts on eliminating some of the more obvious things:
When the Jarduino "chiller" pin activates, does the LED (LED1) on your Relay board corresponding to that relay light up? If so, good, it's sending the signal and the relay board is receiving it. If not, then that could be the issue.
Are you sure you have the polarity correct on the DC side of your AC/DC adapter? That is, with your fan connected to the AC adapter, if you plug it into a working wall outlet, does the fan work? If not, swap the +/- fan wires on the Adapter and test it out again.
Let us know how you make out.
And thanks for sharing your post on the initialization fix. Hope that info helps someone out there.