Super-late to the party here, but after thinking about this, here's a river naming convention I'm thinking about. First a brief history of the various river tiles...
2001: The River (later called The River I). The original 12 river tiles. Contains no additional features with special gameplay effects.
2005: The River II. Has several important differences from the original, including altered road and city configurations, a river fork, a lake with a city that blocks farm wrap-around, a second lake (to support the fork) that also has a volcano, one inn, and a riverside pig-herd.
2006: The Mini Expansion (Games Quarterly #11). Includes 2 river tiles: a revised source tile with a road that blocks farm wrap-round, and a copy of one tile from The River I (a straight river with cities on either side). These 2 tiles are not effectively usable without one of the full rivers.
2014: The River I (Big Box 5 version). A reprinting of The River I with 4 tiles enhanced by either a vineyard or a sheep symbol (there are two of each). This has the same physical river/city/road/cloister configurations, but different gameplay dynamics due to the four added special features. (Thus, players debate whether it should be called River I or River III.)
So here's one way to refer to these that builds little reminders into the version numbers:
River 1 — 2001 original river
River 2 — 2005 River II
River 0.2 — 2006 two tiles from the Mini Expansion, no good by itself
River 1.4 — 2014 river I with four modified tiles
When describing an actual game where you've added the Mini Expansion tiles to a river, you can literally just add the 0.2:
River 1.2 — original river + Mini Expansion
River 1.6 — 2014 river + Mini Expansion
River 2.2 — River II + Mini Expansion
Any game configurations that use two or more full rivers would not add the numbers together, but keep them separate. For example, "Last night we played mega-carc, including River 2.2 & 1.6" would mean the base rivers II and 2014, each enhanced by its own copy of the two tiles from the Mini Expansion.