High vs low internal impedance cartridges....Is there a sonic signature for each type?

The more wire you'll wind the more signal you'll get and impedance will rise. 
Although there is much more between ohms - magnet type/size/form, wire diameter, winding technique, core material (permandur, iron, aircore), size of magnetic gap etc.
So it's better just look only into fingertip - stylus. Life is easier :)

Hi Jim, I have a tendency to like lower impedance cartridges like the Air Tight. The lower impedance cartridges use smaller coils (shorter wire runs) usually made of gold or plated copper. The result is a lower moving mass. I think the only attribute that these cartridges have in common is a higher degree of detail and transient performance. Otherwise, they can still sound quite different from one another. You would think their tracking performance would better but I can not say I have notice a big difference. 

Interesting question.  One would think that the low internal coil impedance group of LOMC phono cartridges would all be lightning fast and punchy, but this is definitely not always the case, not by a stretch.  Detailed, airy and open most often, yes, but sometimes a relaxed sound as well.  It just depends on the manufacturer. 

Nearly all my favorite LOMC cartridges are low impedance, typically 2-3 Ohm. Low impedance cartridges must be used with "current-injection" phono stages designed for low impedance cartridges, this combination is mind blowing. 

The problem is about ''moving mass'' on one side and ''core'
versus ''coreless'' coils on the other. Reduction of moving mass is ''obvious'' by reduction of (wire) windings on the coil former. The lower the number the lower impedance but also the output. To increase the output ''iron core'' is usually used  but has as disadvantage ''magnets saturation''. That is why ''coreless'' coils are used.
This however reduces the output which causes phono-pre problems and /or the sense of SUT's use.

magnets also differs - the most sweet one is alnico, the intermediate is samarium cobalt, the strongest one is neodymium. 
Although if you will look into the picture core moves in magnetic field made by permanent magnet (or field coil but that one is exotic) and less turns makes output more homogenous  but signal/noise factor takes over.  Also most of cheap aircores or in other words coreless coils are bass shy...
But IMO if you will look into core of the problem why we do like old designs fastest/easiest answer is the use of neodymium magnet in cartridge designs from big manufacturers like AT, Ortofon, ZYX, Lyra etc.

Also most of cheap aircores or in other words coreless coils are bass shy...

@bukanova
Very interesting comment, i blamed my phono stages because with all of them (high gain MC or MM+SUT), except just one phono stage with build-in SUTs, my Air Core FR-7fz is a bit "bass shy".

I noticed it depends mostly on phono stage, because in my situation all cartridges are "bass shy" with the same phono stages. And same cartridges have better bass definition with one phono stage i like the most.

air core is bass shy by nature although masters found the ways to compensate that. Ikeda used plastic cube to wind a coils in order to get more homogenous magnetic field and more bass. Friend of mine changed permanent magnets into field coil only due to lack of bass in his air core designs. Aircore isn't the design which is intended to be bass heavy although you can hear bass details  in the music.

Dave Slagle, one of the two brains behind EMIA, has converted a few MCs to "field coil" types, using the Denon DL103, for one example.  I have heard that cartridge at the Capital Audio Fest, in the context of a truly fantastic direct-drive double Quad 57 system, that Dave also designed.  It's hard to tell what makes that overall system so great, but for sure the field coil mod to the Denon seems to be part of the explanation, because I have never heard a DL103 sound so good.  (Chakster, the DL103 may have been re-tipped with an exotic stylus; I have no idea what stylus it bears.) 

I definitely have preferred MCs with low internal impedance vs those with high.  I wanted to point out that this question is really only pertinent to MCs, because MI and MM types are some of my favorites even over LOMCs, despite having relatively high internal impedance/resistance.  I agree most with Nandric's breakdown of the issues that might make low impedance MCs sound best.  Also, besides a "coreless coil" (don't know of any examples of such a type), one might wonder about the Audio Technica ART7, which uses an amorphous core, still has excellent bass response but very low output due to inefficiency of an amorphous core.

Coreless coil the most known example is Denon DL304

and if to speak about mainstream flagman AT-ART1000

Although it’s important to define what does it mean low ohm - to me it’s up to 5 ohms...
in most old SUT’s it’s marked the same, most of them "low " means up to 3 ohms like mine FR XF-1 type L