You misunderstood the result. Whether a flow is possible or not is not relevant. In your first case, where the graph is
$ s \to a \, \ t $
the minimal cut (=partition of the nodes into two sets, one containing s and the other t) is ({s,a}, {t}) which has capacity 0 because there are no edges from one set to the other. The other cut ({s},{a,t}) has capacity 10 (the capacity of the edge from s to a). In the other example with revenue 0, both possible cuts have capacity 0 so you can either do project a or not; both are optimal.
Here is a quite general definition of the terms you're asking for:
A network is a tuple $(G,u)$ where $G=(V,E)$ is a directed graph and $u:E\to \mathbb{R}_{>0}\cup\{\infty\}$. For $e\in E$, we call $u(e)$ the capacity of that edge.
A circulation in the network $(G,u)$ is a map $f:E\to\mathbb{R}_{\ge 0}$ with $f(e)\le u(e)$ for all edges $e\in E$ and
$$\begin{equation}\tag{$\dagger$}\forall v\in V: \sum_{e=(w,v)\in E} f(e) = \sum_{e=(v,w)\in E} f(e),\end{equation}$$
meaning that in any vertex, the same amount flows in and out of that vertex.
Given vertices $s,t\in V$, we construct a network $(G_{st},u_{st})$ as follows: We add an edge $(t,s)$ to $G$ and call the resulting graph $G_{st}$. We define $u_{st}$ to take the same values as $u$ on $E$ and set $u_{st}((t,s)):=\infty$. A circulation in $G_{st}$ is called an $s$-$t$-flow and its value is the number $f((t,s))$.
Now to answer your questions.
- I would go as far as to say that a network really has to be a directed graph.
- A flow is not part of a network, it is something that sortof "lives" on a network. As far as capacities and flow values are concerned: You can have 'no flow' on an edge $e$ by having $f(e)=0$, and you can have 'no capacity limit' on an edge $e$ by setting $u(e)=\infty$. Formally, however, both the flow and the capacity function assign a value to every edge of the graph.
- A network does not have to have distinguished source and sink vertices $s$ and $t$: My above definition leaves that choice open. However, often you want to have something actually flow from some source to some sink - and you picture that source as supplying items by itself, so we want to allow that more flows out of it than in: In other words, we do not want $(\dagger)$ to hold for $s$. We solve this in the above definition by adding an uncapacitated edge from $t$ back to $s$.
I hope this helps a bit, but I suggest reading a good textbook about the topic (i.e. Combinatorial Optimization by Korte and Vygen).
Best Answer
The names tail and head are pretty standard. Better to avoid source and sink because those names are often used to refer to specific nodes in a network, for example, in the maximum flow problem.