Current state: Vanilla
All improvements have +1 yield (except trading posts and GP improvements).
Farms get +1 with fresh water after CS and +1 for the others after fertilizer.
The only other improvement getting a buff is the lumbermill (+1 with steam power).
Current state: TBM
Farms get the CS bonus as usual and ALL farms get 1 additional yield after fertilizer.
Lumbermills and Mines get +1 yield after engineering/machinery. In addition, the LM retains its bonus with steam power.
Trading posts get +1 with economy.
Please correct me if there's a mistake!
Analysis:
In vanilla, tile yields don't change much. During the midgame (from civil service to fertilizer) rivers are even stronger than usual, at least when farming them. The additional yield for lumbermills might mean the devs regarded 3

a bit stronger than 2

1

, so the lategame boost to LMs balances this, making them somewhat equal in average.
I'm not going to explain Thal's motivation for changes, that's best left to him

In general, a production boost is a much requested feature, though there have been doubts recently that the current state might be a bit much. Trading posts mostly receive a boost in the lategame because every other improvement does, too.
Suggestions for "adjacency" concept implementation:
The goal is to make the game more interesting regarding terrain improvements, while keeping Thal's proven balance concepts to a large extent.
1) Rivers and Shores
Rivers in vanilla give 1

for every adjacent tile, regardless of the improvement. I suggest keeping this and expanding it to shorelines.
Historically, both rivers and coasts have been trade routes and preferred settling areas. I see no reason to make a huge difference between them. Also gameplay-wise, especially in vanilla coastal cities are weak, while river cities are awesome. Making coasts similar in terms of extra gold balances the game for civs not starting next to rivers.
In short: River=coast regarding gold (not food, of course!)
This could require sailing, to force players to take at least 1 naval tech, which get little attention usually.
2) Boosts for adjacency
Only farms next to fresh water get a boost for their position usually. I suggest expanding this to
ALL other (major) improvements, approximately at the same time (around medieval era).
The only logical thing that could boost nearby mines is a mountain. I suggest changing the buff for all mines Thal implemented to mines near a mountain. This would make mountains more than just impassable terrain, they would become highly interesting. Production cities would be much better if close to mountain, and civs starting between many mountains wouldn't have a disadvantage any more. The lost yield (compared to TBM, not vanilla) might be a thing to consider, though.
Trading posts and lumbermills are a bit more difficult.
Lumbermills could be more profitable were the lumber can be transported by sea, meaning next to shores and rivers. Lumbermills at shores could also represent ship construction. I'm no expert, but I guess this is historically justifiable.
I tend to see trading posts as villages, so thinking historically they should be around major cities, along rivers and coastlines, maybe also around lakes. Associating them with roads is problematic, though, because we don't want to make roads free again.
Overall, the question is if boosted TPs can be allowed on more spots than the other improvements? IMHO yes, but only if they get 1 base yield like all other imrovements. So basically only TPs in "sweet" spots would get the vanilla yield, although, as I've said, there could be many spots boosting TPs, more than for any other improvement, so 2 yield would be very common. This would also make it more justified to boost them later with economy.
Actually, the number of tiles/areas where an improvement would get added yield could be a way to balance them. Gold is less worthy
per unit than production, food being somewhere in between. So logically, boosted production improvements should be less common than boosted TPs.
Other boosts in the lategame (from fertilizer, steam power, economics etc.) could be added in addition to this system, if necessary.
Questions:
The AI: Since the location specific boosts come rather early, I think the AI can handle it, the benefit the player has because he knows of the added yield earlier should be limited. I'm no AI expert, though.
Automatism: Txurce has mentioned the problem that improvement strategies might be very automated if some tiles give benefits to certain improvements. I think that on the one hand the vanilla improvements are not overly complex, either. On the other hand, since there are many overlaps, you often have a lot of choice. Also, more locations on the map would play a special role, in vanilla mountains and shores have little meaning.
From my experience with Thal's dev version, I strongly believe there would be overlaps where multiple boosted improvents are possible all the time. E.g. a hill 1) between mountain and shore, or 2) city and river: In the first case you have the choice between a boosted mine or a TP, while in the second situation it would be between a hill with boosted TP, or a hill with boosted farm. If the hill is forested, there is even more choice: in both cases the forest would be boosted (through river or coast). As I said above: Those overlaps make the system interesting!
Technologies:
When should the boosts come?
I usually take naval techs very late if I don't
really need them, so I think a non-naval benefit from sailing would make the tech an interesting very early choice. If the gold from river/shoreside tiles would be unlocked with sailing, civs would have to choose it early and would be able to build ships as early as they really came in history. Ships might be important for exploration through this change, too. No longer would remote inland locations be visited by scouts long before the first naval unit explores the coastline!
Civil Service and Engineering/Machinery are fine for food and production boosts.
The TP boost should maybe be on an earlier tech, since TPs are 2 in vanilla and wouldn't be any more until the right tech is researched -> possible lack of gold? Maybe mathematics would be appropiate? If you don't want to crush cities early, this tech is of little use. An useful non-military effect would make archery also more likely to be chosen early.
Multi-Buffs:
If multiple reasons why a TP or other improvement should get added yield apply, the should NOT stack. A TP that's next to coast, river, lake and city should only get 1 extra yield, not 4! Otherwise the choice would be gone again and the whole system gets to complex.
Wouldn't this favor ICS?
There have been concerns about the buff for city adjacency, since more cities would increase the number of "good" tiles for TP construction. This concern is justified. Still, there are ways to counter this.
E.g. a building that gives +1

to TPs but costs 5 upkeep: It would only make sense in cities with more than 5 TPs, and would REDUCE the output of smaller cities. It would only be really useful in cities with 10 or more TPs. This could replace one of the percentage-based economy buildings.
Overall amount of gold and other yields
As I've said, the overall amount of gold should stay roughly the same. The TPs in "bad" positions (no coast/river/lake/city nearby) with only 1 yield would be countered by the additional gold yield for all coastal tiles.
Food would stay the same as in TBM anyway.
Regarding production improvements, while you could build "double" mines and LMs in TBM on every hill/forest, there shouldn't be much less production with my suggestions. You'd only have to make sure you build LMs next to water (for transport) and mines next to mountains, meaning you'd be able to get "double" yield on quite a lot of tiles. This would be a bit less than in TBM now, but I'm sure there are ways to compensate this (lategame buffs?), if even desired.