Does it improve the sound of a MM or MI cartridge by playing it through a phono stage?

Soko, Please re-read my post.  I am afraid you missed the main points.
"You need to match the output of the cartridge with the resistance/impedence [sic] of the phono stage." 

No, you need to match the output of the cartridge with the gain of the phono stage.  The input impedance is also important but only tangentially related to voltage output, in that very low output cartridges tend to be MC types which tend to have very low intenal resistance and can therefore be used with any input impedance, so long as the input impedance is at least 10X the internal resistance of the cartridge.  (Now I opened up another can of worms, not mentioned above. Read the thread about MC cartridges where Elliot is the OP.)  So, it is more correct to say you need to match the internal resistance of the cartridge with the input resistance/impedance of the phono stage.  Gain of the phono stage can never be "assumed to be adequate"; you need to know the value in db in order to decide whether a particular phono stage will work well with your particular cartridge.  For example, a stage designed for MM cartridges (producing about 5mV output) will typically add about 40db of gain in the process of equalizing the signal from the cartridge.  Such a stage would be wholly inadequate for use with an LOMC cartridge producing 0.2 to 0.5mV output.

Then why does a cartridge have a recommended load impedence from a phono stage and so many phono stages have adjustments for that? And why don't I hear about cartridge internal resistance? 

I thought gain was more of a preference as to how loud and possibly distorted you want the signal to the preamp to be so you don't have to turn up the volume as much.

Didn't see a post from Elliott on this discussion.

Look for "elliottbnewcombjr"For example, the thread titled "What SUT?...."
You asked, "why does a cartridge have a recommended load impedence from a phono stage and so many phono stages have adjustments for that?"  Because each manufacturer wants his cartridge to perform at its best.  So he selects a range of phono input resistances that ought to work well with his cartridge and publishes that information.  The rule of thumb is that the input resistance of the phono stage should be not less than 10X the internal resistance of the cartridge (which is another published specification that you should be able to find easily).  Most manufacturers do not stipulate one value; they usually will stipulate a minimum value which allows for higher values of input resistance to be used.  The 10-fold rule is not cast in stone.  You can get away with lesser ratios, but usually you don't want to go far below 10.  The reason for this is that as the input resistance of the phono stage approaches the output resistance of the cartridge, more and more energy in the form of signal voltage is lost to ground.  Also, the high frequency response starts to roll off.  This whole issue mostly pertains to MC cartridges which are often run at phono input resistances much less than 47K ohms.  For MM cartridges, you can forget about it and use the standard 47K ohms, to be found in nearly all MM phono stages.

You may want to do some reading here. I think it will help you better formulate a question so others can understand what you are asking. Lots of great info on all things hi fi.

https://www.audioasylum.com/cgi/d.mpl?audio/faq.html