I will start discussing the relationship between first and third definition.

The key point to understand the different definitions of arithmetic Chow groups is to understand the equation $\partial \bar \partial g-\delta_Z=\omega$. In this equation appears the second order differential operator $\partial \bar \partial$.

There is a cohomology theory, real Deligne-Beilinson cohomology, that contains information about the relationship between the real structure and the Hodge filtration of the cohomology of a complex manifold, where the operator $\partial \bar \partial$ appears naturally as the differential at certain degree. See "Burgos Gil, J. I.; Arithmetic Chow rings and Deligne-Beilinson cohomology. J. Algebraic Geom. 6, 2, (1997)" for a construction of a complex that computes Deligne Beilinson cohomology. It is related to the complex used by Goncharov.

Thus we may write the previous equation as $d_D g -\delta_Z=\omega$ where $d_D$ is a differential in a exotic complex that computes Deligne-Beilinson cohomology.

The next step is to notice that the current $g$ can always be represented by a differential form on $X\setminus |Z|$ that has logarithmic singularities along $Z$. This is proved in the original paper by Gillet and Soulé. Differential forms on $X$ with logarithmic singularities along $|Z|$ allow us to compute the cohomology of $X\setminus |Z|$ with its Hodge structure. Hence real Deligne Beilinson cohomology of $X\setminus |Z|$. From now on we assume that $g$ is the current associated to a differential form $g'$ with logarithmic singularities along $Z$.

If $p$ is the codimension of $Z$, the cycle $Z$ defines a class in $H^{2p}_D(X,\mathbb{R}(p))$ (real Deligne Beilinson cohomolgy). But more preciselly defines a class in cohomology with support $H^{2p}_{|Z|,D}(X,\mathbb{R}(p))$. The class of $Z$ with support in $|Z|$ can be represented in two different ways. With the current $\delta_Z$ or with a pair of differential forms $(\omega,g')$, where $\omega$ is smooth on the whole $X$ and $g'$ is smooth on $X\setminus |Z|$ and has logarithmic singularities along $|Z|$.

The condition $d_D g -\delta_Z=\omega$ is equivalent to the condition

"$(\omega,g')$ represent the class of $Z$ in real Deligne-Beilinson cohomology with support on $|Z|$"

A proof of this equivalence is given in Burgos Gil, J. I.; Green forms and their product. Duke Math. J. 75, 3, 529-574 (1994)

Hence we have replaced a differential equation by a cohomological condition. As pointed out by Myshkin this opens the door to an abstract definition of arithmetic Chow groups: Let H be a cohomology theory that has classes with support for cycles and classes of rational functions with some compatibility conditions. Let $\mathcal{C}$ be a particular choice of complexes that compute said cohomology. Then we can define arithmetic Chow groups with values in $\mathcal{C}$. The properties of the obtained arithmetic groups will depend on the properties of the complex.

This abstract point of view is worked out in Burgos Gil, J. I.; Kramer, J.; Kühn, U.; Cohomological arithmetic Chow rings. J. Inst. Math. Jussieu 6, 1, 1-172 (2007).

Examples:

1) Usual complex of differential forms: The obtained groups are isomorphic to the ones defined by Gillet and Soulé.

2) Differential forms with logarithmic singularities at infinity: We obtain groups with better Hodge teoretical properties.

3) Currents: The obtained arithmetic groups are fully covariant.

4) Forms with log log singularities at infinity: We obtain groups that, on one hand receive characteristic classes from certain vector bundles with singular metrics, namely the ones appearing when studying Shimura varieties. On the other hand they have well defined arithmetic degree.

Thus the main motivation of the third definition is "flexibility"

With respect to the second definition it is a completely different beast. The classical Chow groups has been extended by Bloch to higher Chow groups. This is the analogue at the level of cycles of the extension from $K_0$ to higher $K$-theory. The aim of Goncharov is to construct an explicit regulator from the complex of higher cycles to a complex that defines Deligne Beilinson cohomology, and then use this map to define higher arithmetic Chow groups as the cohomology of the cone of this map. He proves that the degree zero part of his construction agrees with classical arithmetic Chow groups by writing explicitely the cohomology of the cone.

There are two caveats in Goncharov's definition: first it is only defined for varieties over a field, not over an arithmetic ring limiting its usefulness. This is because the theory of higher Chow groups over a ring is not well developed yet.

Second there is an error in the construction. In fact the map $\mathcal{P}(n)$ in Theorem-Construction 2.3 is not a morphism of complexes. This error is solved in http://arxiv.org/abs/1502.05459.