Just as an idea: would it be possible to treat rivers as features? In the same way as forests? That should enable us to have rivers go through the centers of tiles without needing separate "river" terrains. It would enable us to have different kinds of rivers as features, such as "navigable river" or "non-navigable river". And it would enable us, if there is the capacity to add multiple features per tile, to have things like "plains + river + forest".
This is one way to approach it but I don't think it is the best. Whether we use points or tiles (where a point is still at the center), features are centered on points. Rivers would not be. Rivers would be anchored to points and flow along edges that connect points regardless of whether we used points or tiles as the basis for movement and/or placement. Your thought gives me an idea though
One side comment: rivers are not ever considered terrains in Civ 4. Terrains occupy tiles. Rivers occupy the borders of tiles so they are not handled the same way.
In geography/geology,
the concept of stream order (Strahler number) is used to relate stream flow and width to relate that to how many tributaries it has. The smallest body of water (stream) is given a stream order of 1. When two streams of the same number come together, the number is incremented by one. Streams of a lesser order do not increase the order of a larger stream when joined.
So what? Well, we could use a generic version of this system to create the rules for river and tile placement: their placement through or between tiles. Obviously the higher the stream order, the wider it is. Tiny tributaries would not be shown on a map so the smallest rivers would probably of a fairly high stream order. Rivers must be near the highest stream orders in order to support ships. It would be possible to give water based units a "stream order minimum" to designate how small of a river they could handle but for the most part, this number would still need to be fairly high. Even a canoe requires a stream order of 3 or 4 for a smooth journey. It would make sense that only these high order rivers should be bound to go through the centers of tiles since they are the ones that can be navigated by boats and ships.
This paper gives one a good idea of what the Strahler number for United States rivers is.
To give an idea:
Probably the minimum for ANY kind of ship should be 6 or 7.
Cruise and large ships require a 10
Ironclads should probably require a 8 or 9.
Yacht types should require a minimum of 7 and primitive craft a 6.
Honestly though, a map probably wouldn't show a river of anything less than 6 or 7 so we can revamp our original numbers:
large ships: 10
ironclads: 9
small boats: 8
So ideally a stream order of 8 (if you want canoes to travel further) or 9 would be the minimum for requiring the river to go through the center of a tile.