The cases where copyright was denied involved prompt-driven generative AI. At some point the artist admitted that some of the creative decisions were made by the AI.

In your first example, you made all the creative decisions. The same is true of many filters, for example if I apply a "50s cinema filter" then I know exactly what it will do. The AI doesn't get to make any decisions.

On the other hand, if you tell the AI to "add some clouds to the image" and the algorithm decides where they go, then the sky it produces is not protected. Only the elements you controlled are.

At what point does the generative capabilities in Photoshop (which you can give prompts to) become not controlling the output?

At the point where a creative decision is made that the user did not control, that element is not eligible for copyright.

So to return to my example: suppose a photographer takes a photo of a model and asks an AI to add a dog to the image. Unless the dog comes from one of their other photos, they can't copyright it (although they can still copyright the model).

Now suppose another photographer takes a photo of Mt Everest. They ask the same AI to add a dog to the image, which results in the same (or very similar) dog. The first photographer can't sue the second one for copyright infringement, because the first one didn't create that dog and therefore has no rights over it.

At which point does control net (which literally allows you to pose models and replace every single thing about a scene along with numerous other features) become “controlling the output”.

That's like asking "How many alterations do I have to make to an image of the Mona Lisa to transform it into something I can copyright it as my own work?"

The answer is: it's up to the judge. See: Lynn Goldsmith vs Andy Warhol

See any sort of public art, or even the paintings you make by literally just pouring paint on a canvas or hanging a bucket of paint by a string and having it move across a canvas and drip.

An artist pouring paint on canvas or hanging a bucket of paint on a string can still plausibly tell a judge that they controlled the output. The behavior of such mechanical devices is fully predictable by the user. Or they might claim that they controlled some elements of the output but not others. Then they can copyright the elements they controlled.

Of course if they want to lose their copyright, then they are free to tell everyone that they didn't really have any control over the output. See: Monkey selfie.

public art

You can copyright a tangible thing that you personally created but you can't claim a copyright over the interaction of the public with what you created.

To the extent that members of the public contribute tangible creative elements to the work (for example inviting the public to decorate your work), those contributors will share the copyright with you over the final result.