Put another way, you can call things whatever you want, but a marketing campaign does not obligate me to accept your silly branding nonsense.
Conversely, if most people do start calling it East Cut, you will be obligated to do the same if you want people to understand what you mean, since if you'll say "Rincon" they'll (mis)understand it as an area that explicitly excludes territory which they now call otherwise.
I also refer to the Emperor Norton Memorial Bridge, which is my own small contribution to building consensus on naming a different geographic feature.
Bored cynicism kinda requires the underlying reality support one's cynicism.
Call it what you want, but I don't see the point of being concerned over the change.
I get postal mail for someone living a few blocks away because they're technically in another suburb with a slightly worse reputation, and their real estate agent sold them on the idea they were in my suburb instead.
The post office, however, isn't paying attention to marketing or non-profit interest groups, it's paying attention to council zoning and regulation.
So every so often, I get mail addressed to a street name similar to mine, but in the next suburb over.
Names change over time, but when there's a central registry of names with a democratically founded process for managing those names, and the names are a public utility for locating things, it's best we don't just shit all over it and make up whatever we like because it suits us at the time.