I felt that the rules for when you should pay for a river crossing when entering a city was ambiguous so I went digging. I ran across this thread on BGG that helped clarify the costs. I was taught that you needed to look very closely at the position of the river in relationship to where the track enters the city. Although you can make a decision using this method, I don’t think the artwork was intended to be used in this very detailed manner.
If I’m not mistaken, we were paying extra to build track a few times for river crossing when technically we shouldn’t have. The diagram above (created by BGG’s coldark) depicts the widely(?) agreed upon ‘standard’ for the rule interpretation (AFAIK). So, although the artwork may depict a track and a river entering one of the six hex sides, the ordering with that hex side is irrelevant to determining cost. For example, using situation 3 above, assume the river and the track connect with a city in position 6 on the hex. Although the artwork may depict the track entrance on the 5-ish side of the 6, I feel that the track builder would not pay the extra $1000 since both features are within the boundaries of the hex’s position 6.
The bottom two figures depict how, if taken individually, there would be no cost to cross the river. However, if the link at some point clearly crosses the river, then the extra cost would be incurred on the tile that completes the crossing.
This thread has yet another description of the rule that I’m not sure I would agree with. I’m going to stick with the diagram.