I think of a "city bike" a collection of general qualities of a bike, which sounds like what the OP is saying. It doesn't mean it has to be ridden in the city, nor does it mean that any bike ridden in a city is a city bike. It's just a kind of bike, and the description is useful because if you say that something is a "city bike," you imply certain general characteristics. But they are general ideas, not hard and fast rules, so there's a lot of variability in the categories. Likewise my "touring bike" is a touring bike. It doesn't matter that spends most of it's time taking me to and from work. It doesn't matter that I ride it in a city or that it's been known to go off road. If I decided to call it a mountain bike simply because it's been on some trails, it would be very misleading as to the qualities of the bike. And the fact that I ride it in a city would not make it a city bike, but the fact that the handlebars are up relatively high, that it has fenders, that the bars are swept back, rather than drops, might make someone want to put it in the city bike category, and I'm okay with that because they are fairly general categories.
But I agree with the OP that a commuter bike is not a kind of bike but rather a bike defined by it's function. You can call my bike a touring bike or a city bike depending on your criteria, but what makes it a commuting bike is simply the fact that I use it to get home every day.
I think hybrid is just like touring or city or mountain bike: a category of bike that meets some general criteria of features. Some of those do seem like they might overlap with a city bike. In fact, I originally considered a hybrid to be a hybrid between road and mountain bikes, but on further consideration, see very little road bike in a hybrid. It seems more like a hybrid of mountain bikes and city bikes.
I think a certain amount of confusion comes in when people are targeted by bicycle marketing or by bicycle advocacy as being potential commuters. Who are the easiest to try to convince to bicycle commute? People who have fairly short commutes with good roads available to them. In other words, people who could easily do their commute on a city bike. From a language perspective, I think we, especially on this forum, know that any bike can be a commuter, but I can see where some people would think of a commuter as a city bike because it's often marketed as a good kind of bike for a commuter.