I totally agree with Roberto's description of how these rumors and misconceptions get started. Someone who makes one trip to Europe and suddenly considers him/her self an expert, spouts a lot of half-truths and rubbish to an audience that's largely ignorant. The audience then perpetuates these stories and of course sometimes embellishes or changes them.
Back to the water question, in my experience Italian restaurants tend to stock various sizes of bottled water. Some will have both 500 mL and 1 L. sizes, while others will only have one size. I'll usually buy the smallest size possible, but if I have to buy a 1 L. bottle, no big deal as I just take it with me after dinner (or refill my touring bottle).
I've found water in most places in Europe to be safe to drink, but the one exception so far is Greece. There are some places there where it's not safe or not desirable to drink the water. For example, on a visit to Naxos, the hotel staff gave guests a stern warning at check-in about the high lead content of the water (old pipes). Bottled water was readily available at local shops, so not a huge problem. In other places in Greece, it wasn't the safety of the water but rather the taste that was the problem.