Paperw8, before we speak of output levels, we need to distinguish what kind of core the coils are wound onto. Assuming the same level of magnetic field strength (flux), non-permeable cores are more inefficient at converting the physical movement of the stylus into electrical output, but what output they produce is quite linear. Permeable cores still aren't what I'd call efficient, nonetheless they result in much higher output for a given amount of stylus movement, but the permeable core adds noise, distortion and reduces resolution.
OTOH, distortion in a cartridge is caused by physical issues (such as tracking resolution) as well as magnetic, and the better the physical aspects (styli with longer and narrower contact patch, more linear dampers, less body resonances, more complete energy evacuation from the cartridge structure), the more noticeable magnetic and core issues will be. The reverse is also true, if the stylus is conical (spherical) and the damper is a simple one-way design, distortion due to magnetic and core issues will be swamped and therefore much less noticeable.
The next thing that needs to be stated is that distortion or noise in analog playback is caused not only by the cartridge, but also the phono stage and signal cabling. Tonearms likewise cause tracking and energy evacuation distortions, but we will leave them outside of the current discussion (as things would get too complex). In particular, the contribution of the phono stage is not small. Areas that phono stages struggle with include noise and gain (if noise and gain are insufficient, the designer may need to add one complete gain stage or an input transformer), and immunity to high-frequency energy (which are triggered frequently, by ticks and pops, mistracking, electrical loading, and RF issues), which can cause inharmonic distortion, which is particularly nasty-sounding. If the phono stage has a lower level of performance, it is usually better to design the phono cartridge for higher output levels, even if it means so much coil inductance that the electrical phase is seriously messed up (the worst in this respect being MMs).
If the phono stage is state-of-the-art, the cartridge designer can afford to design a lower-impedance, low-inductance, low-output cartridge. The extreme case here would be a ribbon MC (single-turn coil), but to my knowledge, no phono stage has ever been built which could do justice to such a cartridge, which shows you how challenging the task can be.
As far as the cartridge is concerned, lower output is more ideal. Lower output means less metal in the coil windings (copper has a specific gravity of 8-9, which is greater than iron!) for lower moving mass and reduced tracking distortion. Lower output also means fewer coil winding layers, which enables the coils to be of cleaner shape and will improve crosstalk, phase response, and channel matching (cleaner-made coils also look much better).
The majority of cartridges these days have permeable-core coils (well-known exceptions being Benz Micro's ruby core, and the carbon cores used in some of Ortofon's designs). Compared to permeable-core cartridges with their core-induced distortions, air-core cartridges will need more coil windings to achieve similar output levels, and the extra copper may result in more moving mass than if a permeable core had been used, and the increased number of coil layers will impair the geometric shape of the coils. Comparisons between the two approaches tend not to be straightforward.
Feedback from Lyra's markets (we make only permeable-core cartridges) has been that to go below 0.5mV (5cm/sec) means that many phono stages will be less than happy. The user may hear problems like noise, grain, insufficient bass response, or in less problematic situations, they may simply not hear the improvement in resolution that the lower-output cartridge should be giving them.
As a cartridge manufacturer, our problem is that the user may not be happy with the sound, but in most cases they will blame it on the cartridge rather than the phono stage or that they have excessive electrical contact points in the signal cabling system (which seems to work OK with MMs, MIs and high-output MCs), but will impair the sound of low-output MCs. Since no manufacturer likes to hear that users are unhappy, we've shifted our cartridges away from where they were some years ago (0.22-25mV, single-layer coils) to our present level (0.5mV, double-layer coils, although the Dorian and Delos use three-layer coils to generate 0.6mV). We are willing to make single-layer coil versions of our cartridges upon customer requests (especially the lower-volume, more expensive models), and if the customer's phono stage is up to the task, our experience has been that everyone is happy.
Does this answer your question adequately? It's a complex topic, so please feel free to ask more questions if some things remain unclear.
cheers, jonathan