I'm having some trouble finding literature on the developing map.

All the sources I could find on it seem to refer to thurston's definition in either: http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf or http://library.msri.org/books/gt3m/PDF/3.pdf

I don't have much background is geometric topology, so here I'm just trying to test my understanding:

(True or false?): Let $X = \mathbb{C}$ and $G$ be the pseudogroup of biholomorphisms between open subsets of $\mathbb{C}$. Then $\mathbb{C}$ with its usual complex structure is a $(G,X)$-manifold (ie, $(\text{biholo}(\mathbb{C}), \mathbb{C})$-manifolds are just Riemann surfaces). Furthermore, for any simply connected Riemann surface $M$, and any chart $\varphi : U\rightarrow\mathbb{C}$, where $U\subset M$, we have a map $D : M\rightarrow\mathbb{C}$ such that $D|_U = \varphi$.

(True or false?): Let $U\subseteq\mathbb{C}$ be a open disk around 1 on which a complex logarithm can be defined and is invertible. Then, we can view the topological space $\mathbb{C}$ as a Riemann surface via the atlas consisting of the two charts $\log : U\rightarrow\mathbb{C}$, and $\text{id} : \mathbb{C}\rightarrow\mathbb{C}$. This complex structure is equivalent to the usual complex structure on $\mathbb{C}$.

Question: What is the developing map from $\mathbb{C}\rightarrow\mathbb{C}$ relative to the chart $\log : U\rightarrow\mathbb{C}$?

I feel like this can't exist, but I don't see how thurston's definition of (G,X)-manifolds and the developing map manages to avoid its existence.

Another related weirdness is this: In the first link I gave, at the end of page 6, he describes a function $D : U_0\cup U_1\rightarrow X$ piecewise via the charts $\varphi_0 : U_0\rightarrow X,\;\;\varphi_1:U_1\rightarrow X$. However, his description doesn't even give a function on $U_0\cup U_1$, since the transition function $g$ isn't defined on $\varphi_1(U_1)$. It's only defined on $\varphi_1(U_0\cap U_1)$, and so his sending $x\mapsto g\circ\varphi_1(x)$ for all $x\in U_1$ doesn't make any sense. What's going on here?

thanks