Flaws in Lord of the Rings... Yes in the book.

[Elfstone]

I think that the Rings of Power that Gandalf meant were the ones that were directly influenced by Sauron.

You could be right. But when I read the part of the chapter from when Gandalf first starts talking to Frodo about his Ring it seems to me that he's consistently refering to the Great Rings or Rings of Power as being the Nine, Seven, Three, and One. All other rings were "essays in the craft....trifles." But the Great Rings were "periluos." And the sentence before what I quoted above was "A Ring of Power looks after itself, Frodo." Nothing that I see excludes the Three, nor do I get any sense that Gandalf meant to.