How to correctly contract an edge in a network?
The very next sentence (!) of Orlin's paper says


  We point out that the contraction operation may lead to the creation of multiple arcs, i.e., several arcs with the same head and tale nodes.


In some contexts, you might want to replace multiple arcs between the same two nodes with a single arc whose weight (or capacity or whatever it's being called) is the sum of the weights of the original arcs; in other contexts, it might make more sense to keep them separate.  Orlin is keeping them separate.

For example, if the arcs represent trading relationships between companies, it makes sense to merge the arcs: if companies $A$ and&nbsp;$B$ both sell to&nbsp;$C$ and $A$ and&nbsp;$B$ merge, it makes more sense to say that $C$&nbsp;is a customer of the new company once, with a single arc, than to say that it's a customer twice, by having two arcs. On the other hand, if arcs represent roads between towns and two towns merge, it makes sense to keep the arcs separate: if there's a road from each of $A$ and&nbsp;$B$ to&nbsp;$C$ and $A$ and&nbsp;$B$ merge, there are still two roads from the merged town to&nbsp;$C$.  

So, contracting the arc $(1,2)$ into a new node&nbsp;$6$ has the following effects: 


it converts the arc $(1,2)$ into the loop&nbsp;$(6,6)$;
it converts the arc $(2,1)$ into a second copy of the loop&nbsp;$(6,6)$;
replaces the arcs $(0,1)$ and&nbsp;$(0.2)$ with two copies of&nbsp;$(0,6)$;