No, an SBa galaxy has a really big central bulge. We are somewhere between an SBb and an SBc though perhaps much closer to an SBb based on the above representation.
This might now be open for debate for some time. It's funny, we know more about other galaxies than our own in many ways. It's like not knowing what you really look or sound like to others... and then you see yourself on video and are actually are surprised that what you're watching at is really you...
Doesn't it seem mechanically and visually obviuous that a higher rate of collision would result in a bar galaxy, while more slowly colliding/merging galaxies would result in unbarred galaxies?
The barring effect seems to resemble and suggest a high rate near passby by two near colliding galaxies, such that they "rip" partially past one another, and/or merge while continuing to pass one another by, leaving a sorta 'skid' mark between them, eventually merging but the effects remain of the motion afterward.