I think an aquatic fragrance should contain calone, which is the primary defining element. This is the note which gives an aquatic fragrance its watery, ozonic, fresh qualities.
I also perceive calone as producing something of a sensation as well as possessing a characteristic odor. By sensation I mean something like menthol or camphor might produce, but different. Calone has a lively texture, almost a watery, shimmering effect.
Juicy, fruity notes such as bergamot, grapefruit or other citrus, and especially (water)melon and cucumber, are commonly used in aquatics. These notes add interest while augmenting the fresh aspect of the fragrance.
Aquatics usually contain a variety of light floral notes. To me the floral elements are discreet and do not stand out by themselves, but blend together to add complexity and soften the calone effect.
Aquatics usually have very light base notes which do not detract from the overall freshness and lightness of the fragrance. Usually the base notes are some combination of white musks, amber, and cedar or other light woods.
Classic examples include Cool Water (really more of a fresh fougere than an aquatic), Dune, Escape, Kenzo pour Homme, Aqua di Gio, L'Eau d'Issey Miyake, and Bulgari Aqua among others.
I'm not a big fan of aquatics in general, but this is how I define them to myself.