Yes, it is a guitar, of the bass variety.
You make a lot of interesting points, and I see that you are trying to make a strong argument to differentiate the two. You could make an argument that all stringed instruments are different and deserve their own category just as well as you could argue that they are all similar.
An electric bass guitar is pretty much an electric guitar with a longer neck designed to handle the lower strings. I understand how it has four strings and handles the lower registers, just like a double bass, but it is ultimately a guitar designed to do so. The frets make it a guitar. Electric bass guitars were designed to have frets, and be strapped on your shoulder, and to be picked...like electric guitars.
Your point about fretless bass guitars is really where you can start building a case that it is actually a bass. If guitars have frets, by definition, then you can argue that fretless bass guitars are technically no longer guitars.
It would be really great to hear some other opinions on this.