I, too, have searched through Kate Chopin's The Awakening. The best I can guess is that in summarizing Edna's time listening to the music and the influence of the Gulf, that she comes "alive" in some way that is new to her. The fact that "first passion of her life" is in quotation marks may not necessarily mean that it is a direct quotation, but that the author of the summary hoped the quotatio marks would emphasize the birth of new feelings within Edna. The story is called The Awakening for just these reasons, and these examples of music and the ocean give the reader a sense of a striking change in Edna, important elements the author is trying to impart to the reader.
For instance, Chopin writes (in Chapter Nine) about an "abiding truth" that comes to Edna for the first time as Mademoiselle Reisz plays:
It was not the first time she had heard an artist at the piano. Perhaps it was the first time she was ready, perhaps the first time her being was tempered to take an impress of the abiding truth.
The mention of "abiding truth" leads me to believe that this is a step on Edna's part in learning about herself, which I believe is fundamental to the development of her character and the plot, as well as its contribution to the story's resolution.
The other reason that makes me believe that this is simply the summary and a general statement about the things occurring within Edna's mind is the way the summary is worded: i.e., I can find no place where Edna listens to the the music while gazing out at the Gulf. I believe these are two separate incidents that affect her profoundly. The music touches Edna's soul in a new way. And her relationship with the water changes when she learns that she can swim. The water draws her, and it is the water, too, in which she finally finds "release." (Quotations used for my emphasis.)
I hope this helps.