I haven't found much in literature about this process, we can try to find the solution reasoning, but some experiments should helpful.
What is being deposited?
If he connects the positive terminal to the object it means that the object is the anode. So something is reduced on the anode. Copper(II) oxide quite common oxide used for balckening. How can we obtain this by reduction? I would say that it could be the reduction of copper acetate to copper(II) oxide.
So where the copper acetate come from?
In fact, to the cathode there is also a copper wire wrapped around the steel, it could be that the copper ions are produced from that wire on the cathode:
$$\ce{Cu + 2H^{+} -> Cu++ + H2 }$$
Vinegard contains acetic acid and reacts with copper:
$$\ce{2CH3COO- +Cu++ -> Cu(CH3COO- )2}$$
Or directly from the acid attack of acetic acid:
$$\ce{2CH3COO- + 2H+ + Cu ->Cu(CH3COO- )2 +H2}$$
What could happen is the following (decribed here):
$$\ce{Cu(CH3COO- )2 +H2 -> (CH3COO- )2CuH2}$$
$$\ce{(CH3COO- )2CuH2 + Cu(CH3COO- )2 +H2O -> Cu2O +4CH3COOH}$$
Is the deposit related directly to the cathode material?
If this is true there is no relationship with the steel used at the cathode.
What happens if I change the cathode to say normal steel /copper wire?
The cathode may oxidize. If the copper wrapped around the steel has a part on the process, the process could not work if you don't use copper.
Do we not understand the role vinegar and some salt plays?
Salt is probably used to increase the conductivity of the solution.
The vinegard could be used for different reasons, but as I said before it is probably used for producing copper acetate.
Why is this simple process open to debate and conjecture?
This is an excerpt from an article written at the New Zeland Institute of chemistry:
Electroplating is both an art and a science. Although based on several
technologies and sciences, including chemistry, physics, chemical and
electrical engineering, metallurgy, and perhaps others, it retains in
some ways the aspects of an art, in which experience is the only
teacher. In fact, of course, all the sciences have elements of art
which can be learned only by experience; all the reading of textbooks
on chemistry will not produce a chemist. No text on electroplating
will produce an expert electroplater; there is no substitute for
experience and what is somewhat inelegantly termed know-how.
Is not easy to predict the role of each component if the experiment is poorly described, there are many variables.
