Is a consequence what’s missing? It seems, to me, that you have explained the consequence to your daughter, but she is simply too young to understand the finality or gravity of that consequence. I think I’d do what I could to take the decision out of her hands, entirely. And continue to use redirection and gentle reiteration to encourage the behavior I wanted.
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In the example that you gave, for instance....
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Can you move some furniture or collapsible gates in front of the cage so that she cannot get close to them? Put them up higher than she can reach or hang the cage? I realize that disturbing the cage of nesting birds is likely to be just as disruptive as a small child, but maybe there’s some way to shield the cage from her attention? What does she do that’s bothering them? Are they in a nest-box which could be turned to face away from her so that they really don’t see her? Can you cover part of the cage?
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Can you help her understand the bird-boundaries in a more defined way? Maybe something like placing a piece of tape on the ground to show her where she’s allowed to stand in order to look at the birds? Or setting up a “egg-watching” station with a chair and some paper/crayons to draw her observations from a safe distance.
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I’m not sure how much you’ve talked through/built up the impending egg-hatching, but maybe there’s a way to more actively involve her in the care of the birds in a positive way. I.e., “come help me prepare lunch for the birds—they need lots of good food while they take care of their eggs.  Let’s hard-boil some eggs for the mommy bird to eat. Can you refill this cup with water from the sink? Would you like to help me make some new bird toys for the baby birds, once they’re hatched?” My DD often really responds to increased responsibility.
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Can you read together or watch some youTube videos about hatching eggs and discuss in more detail what will happen, why the birds need quiet and stillness now? A lot of my DD’s, um, “exuberant” curiosity can be tamed by an overload of information, to the point where the item in question is no longer quite as interesting.
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I think for me it would mostly be about redirecting her energy and/or encouraging positive interaction rather than just saying "no" a lot.
If you feel as if you must have some kind of parent-imposed consequence when the rules are broken, what about the “time-ins” that are often suggested on this board? If she bothers the birds, she has to relocate to the couch for some quiet time with Mama, some book-reading, something like that?