Duane "Dog" Chapman, what is the word for the part in quotes between forename and surname?
Solution 1:
While tchrist’s answer quite accurately describes in brief what distinguishes various types of nicknames, the specific question here seems to be whether there is a term for the practice of listing a nickname adjacent to or wedged inside a person’s full name. The very brief and short answer is that there is no such term in common use. At least, I had never heard this practice described with a term before researching a bit here.
A bit of research indicates, however, that, although the original definition of the term was somewhat different and relevant only to Latin names in the Roman Empire, this comes under the category of agnomina. The Wikipedia article notes most specifically of agnomina that:
An agnomen is not a pseudonym, but a real name; agnomina are additions to, not substitutions for, an individual’s full name. Parallel examples of agnomina from later times are epithets like Thomas Jonathan ‘Stonewall’ Jackson (though he is known more often by his agnomen than his first name) or popular nicknames like ‘Iron’ Mike Tyson or Michael ‘Air’ Jordan.
On the other hand, the OED definition defines agnomen as:
A name used instead of or in addition to an original or usual name; an agname, a byname; […] [Emphasis mine]
– which would indicate that this distinction is not universal, and that regular nicknames that replace a part of someone’s name are also sometimes called agnomina. (Compare how Michael Jordan is also sometimes simply referred to as ‘Air’ Jordan, where the nickname basically does take over the function of the first name, becoming a kind of pseudonym.)
The original definition of an agnomen in a Latin name is (as defined by ODO):
A fourth name occasionally given as an honour to an ancient Roman citizen
Quoting Marius Victorinus, the Wikipedia page tells us that this fourth name –
[…] extrinsecus venit, et venit tribus modis, aut ex animo aut ex corpore aut ex fortuna: ex animo, sicut Superbus et Pius, ex corpore, sicut Crassus et Pulcher, ex fortuna, sicut Africanus et Creticus
([…] comes from outside, and in three styles, from personality or physique or achievements: From personality, such as Superbus [‘Haughty’] and Pius [displaying the Roman syndrome of virtues including honesty, reverence to the gods, devotion to family and state, etc.], from physique, such as Crassus [‘Fatty’] and Pulcher [‘Handsome’], or from achievements, such as Africanus and Creticus [from their victories in Africa and on Crete])
(The last type is known as a victory title.)
So while the term started out as referring to an extra honorary name given sometimes to certain Roman citizens, it is quite understandable from its description that it would later be applied to our current convention of including various types of nicknames inside (or at least somehow as part of) a person’s full reported name.
Solution 2:
The second set are epithets or sobriquets, because they are things other people call them, more like titles in that they can have separate words and even articles. You would not use a term with an article vocatively; that is, as a form of direct address.
On the other hand, a nickname is usually a short name that you do use to address that person with, such as in Robert Francis “Bobby” Kennedy or in Francis “Frank” Scott Key. If your fellow there is actually addressed as Dog to his face, as in, “Tell me, Dog, how was breakfast today?”, then it is more of nickname than a mere sobriquet.
If your pal James Smith walks in the room, you might say, “Hey Jimmy, wanna join us for breakfast?” You would not say, “Hey (*The) Greek, wanna join us for breakfast?”
Another related but learnèd word is hypocorism. These are little names derived directly from the original, “full” name that are often used on children and intimate, and which may remain with the bearer as they age. This term might be more appropriate in some contexts for those little pet names we call people by instead of their longer names, like Billy for William, Peggy for Margaret, or Paco for Francis.
On the other hand, if her lover always calls Samantha by the name of Kitten, that is more clearly a pet name.
Other related words include bynames, monikers, and cognomina.
Based on followup comments, it is possible that the best word for the OP here is an alias or an assumed name. Those may have criminal overtones, however.
In many cases, no precise distinction can be drawn in the sand demarcating the domain of one of these words from those of the others. They are also used in different registers, since some are French, Greek, or Latin, while others are native English words. Without knowing the fuller context, it may not be possible to favor one or the others.