Yes, it is quite a confusing naming convention. You are right that this is in part an artifact of old video tubes. In the case of digital cameras, this diameter corresponds closely with the diameter of the imaging circle projected onto and around the sensor, such as 1/1.8", 1/2.5", 2/3", 4/3", etc. Here's an example of what this circle looks like:
From the above picture you can see that the diameter of this imaging circle is close to the diameter of the sensor, but a little bigger. This is because image quality drops off drastically as you get near the edge of this image circle, so you want to avoid capturing image information from here if you can. Also notice that this imaging circle has a gradual fade-out; the inch convention for measuring it's width is therefore not a strict measure, but instead represents a standardized/characteristic width (which is useful for comparing the size of one imaging circle to another).
There's one big advantage to convention of describing a camera sensor in terms of its imaging circle: it is aspect ratio independent. The current convention is therefore more a descriptor of the size of lenses required and the impact of aperture on depth of field for a given camera (as opposed to strict sensor size). If we instead used the sensor's diameter or its diagonal, a camera with a 4:3 aspect ratio (such as most compact digital cameras) would have a different size than a camera with a 3:2 aspect ratio (such as most digital SLR cameras)...even if the imaging circles were the same size, and the cameras used the same lenses.
You are right though, many people confuse the two terms. Saying a
camera has a 1/1.8", 1/2.5", 4/3" etc. digital sensor size is definitely misleading, even though this is often times what people say or write. They should instead say it has a 1/1.8" or 1/2.5" *imaging circle*.
Overall though, the size of the imaging circle does roughly correlate with the size of the sensor, so in that sense it is not completely wrong. Only when you are comparing different aspect ratio sensors does it become inconsistent...