There's quite a bit of misunderstanding around this subject. So I thought i'd try to clear the air a bit..
First of all I want to be very clear that casino's are
not able to affect the game in any direct way.. not that anyone will believe me, of course

All games (regardless of provider) run either on the developers own platform, or in some cases (especially with smaller developers) on a third party platform, to which the casino's connect to but don't control. The only real control the casino has is on the players 'Wallet'. Of course, the casino does control the website itself but they have no more control on the games than Netflix has on the content of the movies it distributes, even less actually.
When you press spin, what happens is the game/platform requests the casino to confirm that you have enough funds in your 'Wallet' to cover the bet. Assuming your have enough funds, the casino replies back to the game/platform that it's ok to proceed. The game will generate a result and if you had a win the game/platform will send the win amount to the casino, at which point they will credit the funds to your 'Wallet'.
Of course, to make it more interesting to the players there's lots of bright colors, loud sounds, and things like timing delays to build up the excitement etc.
The entire process is somewhat more complicated but that's the gist of it. Also, for the benefit of this conversation i don't make a distinction between the game and the platform. In this context the two are basically the same but from a technical pov there is a big difference. In this case it doesn't matter so i'm not gonna go into that detail.
But the important thing is that, even though different casinos can offer different RTP versions of the same game (for regulatory purposes games with different RTP would be considered different games and each would have to be certified separately), they are not able to change the RTP on the fly.
Now, about making games with varying RTP...
Some developers indeed make multiple 'versions' of the same game with differing RTP.
The reason is very simple.. to give the casino an option on how much profit they will make from it. There's nothing sinister about this though. As mentioned earlier, casino's operate in different markets that might have very different kind of cost associated with them.. So, even though a casino can make a profit on a game with 96% RTP on most markets, it doesn't mean they can make a profit in all. Or, it could be that they're just greedy.. they're casino's after all, and there's no other reason for running a casino except to make money. But again, it's not like this is done secretly as the RTP is shown in the game.
The most obvious example is the difference between land-based and online casinos. Because the overhead on land-based casinos is higher the games have to be lower RTP for the casino to cover their costs. Think of games that are available on both, like Raging Rhino or Book of Ra.. I'm sure you all know the RTP on land-based is lower but no one cries foul over this.
Generally speaking most casinos choose to offer the highest RTP version to their players. This is an important point for all you tinfoil hatters out there. As, even in cases where casinos are perfectly within their rights to offer a game with lower RTP, usually they choose to offer the higher RTP version. If they were so hell-bent on trying to screw you why would they do this. Again, the reason is very simple... competition. If one casino offers book of dead with 96% RTP and another with 91%, which one are you going to play at? That being said, you should
always confirm the RTP in the game help files before playing.
One thing to note is that making multiple versions of the same game is actually quite rare. Even though there's nothing wrong with it, in reality, most developers will make one version and it's not possible to change it in any meaningful way. Minor graphical changes are possible in some cases. Usually this is done for branding purposes and most commonly with table games... for example, it might be possible to add the casino logo to the chips or to the felt. But, having multiple versions of the same game is rather the exception than the rule.
As for notifying players if there is any change in game RTP.. Like Dunover already showed, the requirements are different depending on what license the casino holds. So, it's worth taking that into account as well when choosing the right casino for you..