The "times zoom" notation is simply the big number divided by the small one, so the examples you give are correct. "3x zoom" simply means the longest focal length is three times the shortest.
This number really isn't very useful, though. On point and shoot cameras, this value became popular in marketing because the widest focal length was generally about the same across all models on the market: they all had a wide-normal field of view. That made the times-zoom a reasonable way to compare how far one can zoom in to get a closer view of a distant subject. The market is more varied now, so that's not so useful.
And with interchangeable lenses, the widest angle of any given zoom can be pretty much anything, so "times zoom" is not useful at all on its own. There is no standard "base" number that the "×" starts from; you go from whatever the widest focal length on that particular lens happens to be. An 18-55mm and a 70-200mm are both about "3x zoom", but a very different range.
On the other hand, the zoom ratio does give you an idea of how much focal length flexibility the lens has, and usually higher numbers are a clue that there will be more compromise on image quality (and/or price, size, and weight).
Photography is a field with a lot of jargon and a lot of numbers to learn. That can be intimidating to would-be photographers who want to concentrate on images, not "tech stuff". A simple number, without any metric-system units, is far less intimidating than needing to learn all about focal length and angle of view, so I don't think the marketers are all wrong to focus on this number for basic cameras.
For interchangeable lens cameras, like digital SLRs or mirrorless compact system cameras, in some ways the complexity of using focal lengths is a selling point. Intermediate and advanced users may prefer to be given the straightforward facts instead of having to decode more-removed numbers like times-zoom. In some ways, giving the angle of view instead of focal length might be preferable, but that hasn't really caught on — probably because it's really not very hard to get a sense for what different focal lengths mean for field of view on your own camera, once you get over the initial learning bump.