As a practical matter, I tend to agree that Bob is a better descriptor of the instrument we play
As an academic debate, I would side with calling it a bass guitar. Its form evolved in the guitar family, adapting functional characteristics from the double bass. The form aspect of an instrument (design and construction), in my view, is what makes it a member of its respective family of instruments.
While the guitar and the bass share form, there are functional differences that often require different technical skills. However, I do not perceive this as sufficient to overcome the form similarity between the two so as to state that the bass is separate from the guitar family.
Perhaps this may be an inadequate analogy, but I think that guitar is to the bass (not the double bass) what the violin is to the cello (violincello). The violin and the cello are classified as viols. However, though the violin and the cello share similar form, they require different technical skills (bowing, plucking, fingerboard technique). Additionally, the violin plays the treble clef, whereas the cello plays the bass clef. But despite their functional differences, their form is essentially the same, hence they are both viols.
This is how I perceive the guitar to bass relationship. The guitar evolved over time, adapting and borrowing from other string instruments (lute family). At some point, it became sufficiently distinct in form to be differentiated as a guitar. The bass follows the guitar form. It requires different technique from the guitar, and is written in a different clef, but the form dictates they are both guitars.