I found a paper describing a buoyancy engine that specifically notes an external reservoir. I did not find an "official definition," however. In the past, I have attended an autonomous watercraft competition, the name of which escapes me. One of the exhibitors in the display tent promoted water glider kits for students. The description you've provided regarding the syringes is nearly identical to the mechanism used in the water glider kit.
Pulling in water from outside into the "engine compartment" results in an increase in weight as well as an imbalance based on the placement of the water portion of the syringe. The resultant sinking nose down will cause the glider to move forward (via wings) until the desired depth is reached.
Pushing the water out increases buoyancy, again at the nose, creating a climb and forward "flight" in the water.
Lacking an indication of "official definition," I would suggest that what you describe and what I've seen qualify for a buoyancy mechanism, if not a buoyancy engine. Propulsion via buoyancy seems to be the correct assignment of the terminology.
I'm not quite sure why the external reservoir is part of the system. External to what? Does it float on the surface or is it secured to the bottom of the ocean?