Hello Axel . . . consideration of some of the specific requirements of MC vs. MM cartridges was something I spent quite a bit of time with in my own phono stage design, and I very much feel that the usual "change the gain and loading" approach cannot begin to deliver optimum performance from both types. And while my design is specific to lower-output MCs, I designed, measured, and experimented with some MM-specific approaches.
The most important part IMO is the specific noise/impedance relationship between the the input stage itself (NOT just the loading), and the cartridge. MM cartridges have much higher and much more reactive source impedances than MC cartridges, which means they will require an input stage with lower noise current (hence higher noise voltage), than an MC cartridge, which is exactly the opposite. JFETs and vacuum-tubes are the traditional choices here for such an application, though a lightly-biased bipolar can sometimes work well too.
The second input-stage issue for reactive sources like MMs is capacitance, which can be static or dynamic (changing capacitance with signal level), the latter of which will have a big effect on the ultrasonic behavior of the cartridge. Vacuum tubes are pretty good here, as their input capacitance is usually mostly static (like cable capacitance) for the signal levels we're talking about. With JFETs dynamic capacitance is a big issue, and cascoding a JFET input stage is IMO mandatory to keep this under control for an MM cartridge. Bipolars have such a high transconductance that the capacitance is rarely a problem, but they in turn require special attention to input-bias current so a tiny DC current isn't drawn through the cartridge itself.
For the devices themselves, I think it's silly to turn one's nose up at modern monolithic opamps, especially if a JFET input is what's decided upon. In the past 15-20 years, most of the innovations in JFET fabrication have occurred as part of improved monolithic processes, to the point where JFET opamps can be had that beat the noise performance of the very best discrete JFETs available.
The other question is whether or not to have a balanced input from the cartridge, or a simple unbalanced input. This is even a separate question from whether or not a differential-amplifier is used as the input pair, as the latter is frequently used simply for the feedback capabilities from an unbalanced input. For MC cartridges, I think a balanced input is optimum, especially with a transformer input - this is in part because the low source impedance of the cartridge can make the whole system (cartridge, cable, and input stage) very effective at eliminating magnetic hum pickup.
But the balanced input almost always means a noise penalty of 3-6dB when implemented with active electronics (transformers are unsuitable for MM cartridges). And I'm not sure that with the higher/reactive source impedances of MM cartridges, combined with the low-capacitance cable that they prefer . . . really translates into improved real-world noise rejection when used with a balanced input. At the very least, it greatly complicates some of the decisions required to make the input stage tolerant of common-mode noise, and provide proper protection against accidental overload (30VRMS common-mode nose from a bad turntable ground, anyone?)
Anyway, these are just a handful of the overall-picture considerations that can be very different between a phono preamp design that's specific to MM cartridges.