Thinking my Flonase (Fluticasone) nasal spray bottle was empty (which is actually was, I have no idea what suddenly changed), I started pumping it (which I usually have to do when it's empty) before I reached my nostril, and a puff sprayed into my left eye. As I heard the sound, I immediately closed my eye, and then rushed to the sink and rinsed it. I continued to rinse it and even made an "eye bath" out of a shot glass. Though the capillaries in the outer corner of my eye are slightly inflamed,

It sounds like you reacted quickly by flushing your eye, which is great. A slight inflammation of the capillaries is common after an accidental spray, but if you experience prolonged discomfort, redness, or vision changes, it's best to consult an eye doctor. Keep monitoring your symptoms, and avoid rubbing the eye to prevent further irritation.