When water is distilled, the purpose is to remove metals and other contaminants; doing that actually sterilizes the water. The bottling and sealing process may or may not be done in sterile conditions.

Distilled water used in medical settings is bottled in sterile conditions, labelled sterile, and provided in ampules, bags or bottles of appropriate size to be completely used in one application. Leftover amounts are not considered sterile and are discarded (for example, a squeeze bottle used for wound irrigation is tossed after the wound is bandaged.)

When distilled water is bottled for commercial use (like the gallon jugs in the grocery store) they are not labelled sterile, so the bottling process is unknown. And once the bottle is opened, airborne contaminants can find their way into the bottle, so it can no longer be considered sterile. If the seal is compromised during handling, it may or may not be sterile when you buy it. That said, distilled water is nutrient-poor, so it is a poor growing medium for bacteria, but...

I don't use any sinus rinse, but if I did, I would err on the side of caution and boil whichever water I use according to their instructions.

As for Crystal Geyser being the recommended spring water, I believe it is a matter of personal taste and availability. Have you found something different?

Sue