Background
The manual drip technique (AKA "filtercone brewing") and an automatic drip-brewer are very similar; both involve pouring water through a conical filter into a vessel. The difference is that an automatic drip-brewer maintains consistent and hopefully ideal conditions, so when you drip-brew manually, you are essentially trying to recreate the environment inside an automatic drip-brewer.
Much of the following is derived from sweetmarias.com - my constant source for coffee information.
Customizing your coffee
The most important variables in terms of how much they affect the end result also happen to be easily isolated variables that you can control simply with tools. They are:
Temperature of the water
The best temperature is when the water is just coming to a boil, around 200° F or 93.3° C. An automatic drip-brewer will generally maintain a temperature of between 195° F and 205° F.
Water that is too cold won't extract enough of the oils and will give you flat, flavorless coffee. Water that is too hot will extract a lot of the proteins and give you very bitter coffee.
This is simple to control for; just use a thermometer, or learn how to recognize when the water is just about to boil.
Coarseness of the coffee
The grind (obviously, a fresh grind especially) is important here, as it is with any distillation method. What you need to remember is that you are matching the temperature and pressure of extraction.
For high pressure extraction (espresso), uniformly fine grind is preferred to ensure that the water does not shoot past the puck too quickly. For lower pressure methods (i.e. manual drip), you want a medium grind to ensure that the water passes easily but not too quickly.
Again, this is simple to control for; just use a good, reliable coffee grinder.
Ratio of water to coffee grounds
This is mainly a matter of personal preference. Figure out what you like, but ensure that this is the final variable you are isolating. Use a measuring spoon, start at 8 grams (about 1 rounded tablespoon) per 5 oz water, and start experimenting from there.
Improving your consistency
Once you've dealt with the obvious factors above, proper technique becomes essential to improving on the consistency of your brewing process. You'll want to watch out for:
Steady heat during the pour
Depending on the material your filter cone is made of, it may take a while to heat up, which means that initially the water will lose heat to the cone and become too cold before it passes through the coffee.
You can compensate for this by pre-heating the cone. Use hot water for this; don't use the microwave, especially if it's a plastic cone.
Ceramic cones are more heat-stable than plastic, but will take longer to heat up, so it's more important to pre-heat them. You probably don't need this stability because the water shouldn't be in the cone for too long - unless you're brewing a full pot. Most pour-overs should really only be for about 20 ounces of coffee.
Pour Time
The best automatic coffee makers have a sprinkle head and pour out over a deliberately extended period of time, as opposed to shoving the coffee through in a rush. This is because you are trying to extract flavor as thoroughly as possible, and quick is not thorough.
Maintain a slow, steady pour. Trickling is less effective since it will cause an uneven leaching from the grinds, so be wary of this. Submerging is even less effective because you will lose heat from your water as it sits there. This is probably the most important part of a good brewing technique.
It should take 4-5 minutes, pouring evenly over all the grinds, to extract a full 10 cup pot. Hopefully your arm is strong or you're only doing 1 cup.
More practically, you will want to refer to this chart on specific infusion times: For example, for a #4 filter, and 20 ounces of coffee, you will want to spend 2 minutes and 30 seconds on the infusion (they recommend 32g or ~4 rounded Tbsp of coffee).
Wetting the filter
This applies when your filter falls down (as wetting it keeps it up) - not a big problem with conical filters. That said, in some cases rinsing off your filter is preferred in the event that your filter gives an off-flavor. I personally have never encountered this with unbleached filters.
Wetting the grounds
There are two schools of thought on this. Some say to do it, others say not to do it because it locks up the oils or stales the coffee when done too far in advance.
In the case of a straight pour over filter, non-vacuum, sweetmarias.com recommends "blooming the coffee 15-30 seconds" by sprinkling with a pre-pour. (I assume that's what 'wetting the grounds' meant. In my experience, premature moisture is less of a problem for beans with more sugar, i.e. lighter roasts.)
As for time between wetting filter/grounds, you should minimize wetness of coffee in the interest of freshness by thermal conservation, or refrain from either step.
Height of pour
This matters mainly because you don't want to splash water or grinds everywhere, but more importantly you don't want the water to lose heat on its way down by pouring too far.
Rotation of pour
Pick one, do it evenly across all grinds. The important thing here is consistency.
Miscellaneous tips and tricks
According to sweetmarias on the quantity of coffee:
Simply using more coffee grinds cannot fix other brew problems: If you use 20 grams coffee and 350 ML of water and 4 minutes steep time to achieve 20% extraction (it should), using 40 grams coffee with a contact time of 1 minute to compensate will not result in a better cup.
Consider removing the drip before the final drops go through. According to Voilley et al., Eval., 287:
Perceived coffee bitterness is lower when coffee is brewed hot than when cooler water is used. This is hypothesized to be due to the heightened aromatics released in hot coffee, which counteract the bitterness.
So watch out for those last few drops, because they will tend to have lost some heat and will thus leach more solids with fewer aromatics, resulting in more bitterness.
Now go enjoy some hand cranked coffee consistently.
Assuming you're talking about USA usage, you're correct, a "cup" is usually 6oz. In the USA, the standard size for a "cup" of coffee is 6oz, even though nobody drinks cups of coffee that small (12oz to 20oz is more common). For that matter, the size of a "cup" of tea can be 5oz or 6oz when the number of "cups" a teapot holds is listed; a "6 cup" teapot is only 32oz.
However, be careful how the word is used in American recipes. If a recipe calls for a "cup" of coffee, they are more likely to be calling for an 8oz cup, rather than a 6oz cup.
I can't find a clear reference as to where the unrealistic 6oz measurement for a cup of coffee started. Possibly the result of Mr. Coffee, but we're stuck with it as customary now.
However, even though a 6oz cup of coffee may be customary, specific coffee maker manufacturers may use different measurements on different models, including cups as small as 4.2oz. So don't assume unless you've checked.
Oh, and also note that the 8oz cup is American, rather than Imperial measurement. An Imperial cup is around 10oz, although you're unlikely to encounter this measurement in any recipe published after World War I.
Confused yet?
Best Answer
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).