My dad told me mountain spring water may just be tap water in a bottle
Even if that was the case, most tap water is municipally filtered. Certainly bottled water, if it was out of a tap, would be filtered. The last thing a company wants on its hands is a lawsuit.
I had this exact fear in August of last year. I was in Collingwood, ON, on vacation, and I got a severe amount of water up my nose from a fairly unclean public swimming pool. The next ten days were hell for me, and I kept counting down my life like it was about to end. I was obviously fine since here I am talking to you, but I guess the lesson is, even in clearly unsanitary cases like a lake or an unclean or under-chlorinated pool, how many people swim each year?
Millions in the States and Canada alone.
And there are 0-8 cases per year. If I remember the statistics, 8 cases happened only once. Usually it's no more than 4 cases. Among those cases, there is usually some degree of negligence. Swimming in a freshwater lake that is dirty, or using well water in a neti-pot without boiling the water for ten minutes first. Even among those dirty lakes, millions swim and, like 4, actually get N. Fowleri.
You literally have a better chance of getting hit by a bus if you were to walk outside right now and cross the street. There's about a paragraph of information in any given medical journal (believe me, I've looked), whereas there are sometimes dozens or even hundreds of pages or entire books dedicated to other illnesses.
I guess my bottom line is: you're fine, like all of us, including me, who have had this fear. :)