Natural Black Diamonds are actually quite different than other colored diamonds. The "dry" explanation to what a black diamond is, is an impure form of polycrystalline diamond consisting of diamond, graphite, and amorphous carbon. Not that clear...
But in other words, while "regular" natural colored diamonds get their color due to impurities that are attached to them during the formation process (for example boron causes blue diamonds), black diamonds are similar to white diamonds that have an extremely high amount of inclusions, clusters of graphite inside of them to the point that they simply appear black.
Treated black diamonds are regular white diamonds that are usually of extremely low value due to the high amount of inclusions in them. Since they have so many inclusions, as white diamonds the only use for them is as industry grade diamonds, however, with the help of treatments such as irradiation or heat.
In a way, a more suitable name for these diamonds (which I've seen used numerous times) is "colorless treated black diamonds" or simply "black colored diamonds". Since for the creation of this type of black diamonds "useless" white diamonds are used, these are the cheapest type of black diamonds.
Man-made / lab grown / synthetic / Simulant black diamonds are not diamonds in the normal sense of the word. While to me these are basically all more less the same, I believe that those who deal with these type of diamonds would not agree with me (and would be very angered with me).
All of these are usually referred as faux but to be honest there is a vast difference between them. But, to be honest, there is a vast difference between them.
Lab grown diamonds are actually, in a way, hi-tech companies and while treated diamonds are vastly cheaper than regular diamonds, lab grown diamonds are still not as cheap. The reason is that there are tons of research expenses, heavy and expensive machinery, men power.
This is why while you can see a market for lab-grown blue and pink diamonds there is no market for lab grown brown diamonds or black diamonds - there is no economic sense. Natural blue and pink diamonds cost tens if not hundreds of thousands per carat and brown or black diamonds cost about $2,000 - $3,000 per carat.
The reason I did cover the subject of "man-made black diamonds" here is because I know it is searched for and written about but to be honest I have never encountered such - probably because it is not economic to produce. Also, as other hi-tech products, with the advancement of technology their prices are constantly going down.
Simulants / synthetics on the other hand are just like buying a cubic zirconia and are almost free - you can buy them "a dime a dozen" in sites like Alibaba. You should not buy them if you are looking for black diamonds but you should know about them and watch out for them.