Fan123 said: "I interpret the title song of 'The Light in the Piazza' to mean that Clara is likening her love for Fabrizio, to light itself. Most people probably can't rationally explain or understand light (I sure can't; photons?!) but they can still see by it just as well as people who understand light better. I think Clara is saying that similarly, she may not have the intellectual capacity to explain her love or even completely understand it herself, but she can still feel it just as deeply as anyone else, and so it's still valid. (Whether she would have the capacity to make such an argument is another question, but I like that she's given some voice in the show to advocate for herself, whether it's strictly realistic or not.)
And, of course, 'Clara' means 'bright', so there's also that."
I agree with your interpretation. I think that the light in the piazza is something tangible, familiar and known to Clara. She appreciates its beauty (Tiny, sweet and then it grows and then it fills the air). Her sexual and romantic awakenings with/for Fabrizio are new and unfamiliar, however. Twice she sings, "Who knows what you call it, I don't care!" and "Who knows what you call it, but it's there!'.
Towards the end of the song she proclaims her realization, "I see it! Now I see it!". She ends the song by very simply equating the two (she doesn't even use a complete sentence to do so): "The light in the piazza... My love..."