I apologize for the delay in getting back to you all, I've had a very busy few months. I ended up buying both the P180 and the Define Mini so I could do an in-person comparison on noise levels, and ultimately I decided on the Define Mini over the P180. Both of the cases were very quiet, but the Define Mini simply blended in to the background noise when fully loaded with hard drives and fans, whereas it was always possible to hear the Mini P180 when in use, however slightly. I am being fairly nitpicky however; both of these cases fully loaded were substantially less noticeable than the buzz from my LCD monitor's power brick (which, I will admit, I can only hear because I'm young and have sheltered my ears from loud noises my whole life - no rock concerts, loud bars or subway rides without earplugs!). To anyone who was wondering, 99% of the smell from the Define Mini dissipated within an hour of opening the case, and I now cannot detect any foul odours from the sound insulation or the rubber grommets unless I stick my nose right on them.
The Mini P180 was very solid, but despite its build characteristics and silicone mounts for the hard drives, the Define Mini had much less pronounced seek chatter and 'whirring' from the hard drives - I attribute this to the Mini P180 having an open top to the case (for the 200mm fan), whereas the Define Mini has a solid top and a cover over the vent. The Define Mini did have a bit of an echo problem with the Corsair H60 I mounted (buzzing mechanical noise reverbrating off the roof of the case), but this was solved by running the pump at the 'auto' fan setting instead of 100% (temperatures are the same under load testing, so there is no downside to running the pump slower). My only complaint noise-wise with the Define Mini is that the top panel of the case is not fastened to the chassis as well as it should be, and it rattles substantially if you tap it. It does not rattle from hard drive vibration, however. To fix this problem in future revisions of the case, I would add two more rivet points on each side of the panel towards the middle to better secure it to the chassis. Additionally, adding sound dampening insulation to the roof of the case would be a much appreciated change, although based on my experience it isn't strictly necessary.
However, both of these cases were very good for my use. I did run in to one problem, however: the silent Gelid fan I have been using straight off the motherboard apparently has an incredibly high startup voltage. The two 120mm fans that Fractal includes with the case will start with the included fan controller at the minimum fan speed setting, but the Gelid will not start at anything less than 80% fan speed. This is a minor annoyance; I just crank the fans to max when I turn the computer on, then turn them down to minimum to keep all three fans running silently. I will likely look in to a low-voltage startup silent fan in the future, but for now, this is quite satisfactory.
Tl;dr: both the P180 and Define Mini are good to great, both have minor quirks that could easily be fixed by the manufacturer in future revisions, and both were very unobtrusive when fully loaded. The Define Mini 'aired out' quickly. I chose the Define Mini for my purposes and sold the Mini P180 to a friend building a small server.