--OK, The main distinction that I want to draw first is based on the way the cardioid pattern is obtained. There are two general approaches to capsule design for cardioids, both of which run into fundamental problems starting around the resonant frequency of the system. But the older approach also has a real problem at low and low-mid frequencies, while the other, more modern approach solves that problem nicely.
The thing is, not everyone wants that low-frequency problem to be solved. Some people may not even see it as a problem, based on the way they use microphones. I think that people who use cardioids for coincident or closely-spaced stereo recordings need to see it as a big problem, however (see explanation below).
So: I'd like to use as my examples three "vintage" Neumann microphones that are well known among studio engineers. And by the way, when I say studio engineers, I don't mean to imply any particular level of professionalism or technical awareness; I just mean to differentiate the majority of their type of work, and their typical microphone usage, from live, on-location stereo recording and the way we use microphones in that kind of situation. Microphones that might be the greatest in the world for one type of recording might not be very high on the list for the other.
The first, and historically earliest, of the microphones I want to show you is the U 47. It was introduced after World War II, but its capsule (the "M 7") was an older design, having been introduced ca. 1932 as one of the options for their modular system at the time (using the CMV 3a amplifier). 80+ years later, both Neumann/Berlin and Microtech Gefell still manufacture the M 7 in updated versions and sell microphones based on it. It's a dual-diaphragm capsule with a shared backplate. Some of the holes in the backplate go all the way through, so that the air cushions behind the two diaphragms communicate with one another--the so-called "Braunmühl-Weber" capsule type. (Dr. Hans Joachim von Braunmühl and Walter Weber of the Reichsrundfunk-Gesellschaft published a highly influential textbook on applied acoustics in 1936; this type of capsule is shown on page 56 of their book, the relevant part of which is shown below. I'll translate the text as a P.S. to this message, since it's interesting in a number of respects.)
Anyway, if you look at the midrange polar diagrams you see something that's not quite cardioid--there is a tendency toward supercardioid. If you go upward in frequency you see a distinct narrowing of the pattern, consistent with the size of the diaphragm (slightly above 1" diameter). But the main thing I'd like people to notice is what's happening at mid-to-low frequencies: The diagram looks like what we would call a "wide cardioid" pattern nowadays.
In studio applications (e.g. solo vocals or spot miking of an instrument), the narrowing at mid-to-high frequencies gives the microphone extra focus on whatever it's pointing toward. (Its 0° frequency response also includes a generous "presence peak" as shown.) The extra pickup around the sides and back at low and low-mid frequencies means that more room sound is included, giving increased depth, warmth and roundness to the sound. This is one of the most widely imitated studio condenser microphone types.
But it would be a horrible choice for coincident or near-coincident stereo pickup, because this spreading out of the pattern at low and low-mid frequencies would make the recording nearly monophonic in that range. The sense of spaciousness in a stereo recording depends critically on preserving the full difference information between the two channels at low frequencies. So this is a clear example of a microphone that is greatly admired in the one realm but nearly useless in the other. (If you enjoy the boosted lows and low-mids, my suggestion would be to record with microphones that have a cleaner pattern and then apply EQ to your liking in post-production. The stereo effect is much better that way.)
By the way, the U 47 was a switchable-pattern microphone; if its rear diaphragm was polarized at the same time as the front diaphragm, it would pick up sound from the rear as well as the front in like polarity. I wouldn't exactly call the resulting pattern "omnidirectional" because of its substantial irregularity, but it was used as an omni sometimes, and it was definitely better to have the option than not to have it. -- In the late 1950s Neumann introduced a variant of the U 47 called the U 48, using the same capsule; it had switchable cardioid and figure-8 patterns. But capsules for the U 48 had to be specially selected for symmetry between their front and back faces, so that the figure-8 pattern would have its null at 90° rather than being skewed one way or the other.
--best regards
P.S.: Rough translation of page 56 (the German seems pretentious and tiresome even to me): As was indicated in section c for the dynamic microphone principle, a microphone arrangement with a one-sided directional effect can also be achieved on the capacitive principle through a combination of the capacitive pressure-gradient microphone with a normal condenser microphone. (Note both the use of "normal" to mean what we would call "omnidirectional" today, and the use of "microphone" where we would say "capsule". The capsule referred to is a large-diaphragm pressure transducer, which because of its size, is omnidirectional only at low and mid frequencies. Its pattern begins to narrow in the upper midrange and it ends up being quite "beamy" on top. Maybe that's why they didn't call its pattern omnidirectional or "spherical". --DS) This is especially simple to carry out with an arrangement as shown in figure 50. Sound pressure causes two force components to affect the two membranes. The sound pressure produces a membrane displacement that corresponds to the change in the enclosed air volume; the two membranes may move toward one another or away from one another. On the other hand, the pressure gradient produces a membrane motion in the same direction, and thereby a parallel displacement of the contained air volume without significantly compressing it. By means of a suitably dimensioned microphone body (again meaning capsule; from here on I'll just write capsule without noting it --DS), the displacements caused by both effects can be made equal to one another. When both components of motion occur in the same direction, as when a sound comes from the front, the displacements add. But when a sound comes from behind the capsule, a quantitative compensation occurs; the capsule's sensitivity reduces to null.