You list the main differences already (and yes they can be both made from the same bean or blend).
The name turkish coffee refers to the preparation method ... and the grind/granulation/coarseness is adjusted to the method.
Coarse ground coffee - Turkish preparation:
I come from a culture where Turkish coffee is brewed in almost every
home, so I always preferably go for it. And sometimes not having the
fine turkish-style coffee powder at hand (living abroad) I have made (read: tried to make)
"Turkish coffee" with a coarser ground coffee powder many times -
the taste is often quite alright, as you can compensate for the
surface (exchange area) with adding a bit more powder or letting it
sit a bit longer. However, as for this preparation you leave the
powder in the coffee (also while drinking) the coarser coffee does
not sink easily ... so you get it all between your teeth, if you
don't filter it out somehow (which is then not the traditional
Turkish coffee anymore).
Turkish-ground coffee - filter preparation / French press:
On the other hand using a French press, the fine Turkish-ground
coffee escapes through the mesh. This is not too bad, as the super fine coffee will sink and you will end up with coffee brewed in the carafe (like the "mud coffee" or kafe botz that they drink in Israel), with the unnecessary addition of making the French press dirty. Used to prepare filter (drip) coffee, the
finely ground coffee clogs the filter paper. (being the adventurous
type I have experienced all this already, but wouldn't dare
recommending it)
In both cases you get coffee as a result after some hassle ... and even the desired taste might be reached ... but using the right coffee grind with the right method makes everything way easier.
The best is if you can grind your beans yourself (in Europe you can find public coffee grinders in many supermarkets and stores) ... then you can always achieve the desired coarseness for the right method of preparation (drip brew, french press, espresso, turkish,...).
Some additional notes:
You mention arabica - it is the most widely cultivated and
commercially used species of coffee (Coffea arabica) - so you will likely find it used in most preparation methods; the
other species that you sometimes get is called robusta (Coffea
canephora) and has a much stronger bitter flavor.
Not only the specieas, but also the roasting is very important for the flavor. So with the same bean you can get very different coffee (from a very strong dark roast to a very light roast). In some places I have seen turkish coffee prepared by quickly roasting the dry coffee powder in a cezve before boiling water was added, which gave it an extra toasty aroma.
In some places, turkish coffee will be made with the addition of spices (e.g. cardamom), which you can find also already in some ground Turkish/Middle Eastern/Iranian coffee blends. This will quite likely make a big difference in flavor when compared to "French" coffee (or to some drip-coffee blends that are sometimes made with the addition of hazelnut or vanilla flavors).
In short, it's about extraction. Too fast, and you don't get the right flavours and too slow and you'll get off and bitter tastes.
With a setup like the one you describe you can't change the speed the water passes through coffee (unlike espresso, for example) so you control how long water spends with your coffee (and the surface area of contact) by the size of the grind.
The water will tend to stay and mud up super fine grinds. If you were to try to filter through turkish grind, you'd be waiting a long time and might taste something harsh.
Too coarse of a grind and the water will run through and not pick up enough (low extraction).
Aside from colour, taste and aroma, what you are looking for is how long it takes for water to pass through the coffee grind. I can't tell you how long for your specific setup, but if you're going slower than 20 seconds per ounce of water, it's likely too fine of a grind. Also note that the longer water spends with coffee, the more caffeine it'll pick up.
Finally, if you can, keep an eye on the drips as they leave the filter. When the drop or stream is no longer cloudy and starts looking clear, that's about when you've extracted what you can with that grind (both amount and coarseness).
Best Answer
This is something that is likely to depend somewhat on your preference. That said, when I cannot get purpose-made turkish coffee, I usually use a very dark roast for turkish coffee, mixing it with a bit of cardamom (about one tablespoon to every twenty tablespoons of powdered coffee).
From the offerings you have available, I would recommend the Very Strong "Arabica french roast". You may very well find that this is too bitter for your palate. If so, I would recommend sweetening the coffee using sugar, rather than going for a lighter roast, but again, you may find a lighter roast.
If you have one of these roasts in your house already, I would recommend starting out by trying that, then playing around with spicing it with cardamom, and sweetening with sugar before getting a different type of coffee. This will help you determine what way you want to go with your finished brew.