I'd say 3 D-cell batteries is perfectly correct, although technically annoying.
At the most basic level as mentioned, a "cell" is a chemically active construction capable of delivering current with a voltage differential. It could be a pool of chemicals with anodes, cathodes, separators, etc, but this wouldn't be very useful in application...
A battery (in my opinion) is the useful combination of cell components into a coherent self-contained portable (definition of portable may vary, but something that can be safely transported by some physical means without excess care taken (for instance, no worries about sloshing your chemicals out of their vats)) package which is usable on it's own or in combination with other batteries. Each battery may consist of multiple cells, or be a single cell packaged on it's own, so long as it's usable independently.
A collection of independently usable batteries, packaged together in a fashion which can be separated without permanent damage to any of the individual batteries is a battery pack (for instance soldering tabs together and shrinkwrapping, can be unsoldered/cut apart and shrink wrap sliced open, and the batteries (of one or more cells each) can be used freely).
A collection of batteries permanently joined together in a fashion that cannot be non-destructively separated I would also consider a battery, not a battery pack. For instance if you take 2 marine 6V lead acid batteries, and use a plastic welding glue to mate them solidly together, that is now a 12V (or 6V 2x capacity) battery on it's own (consisting of 6x2V cells, or 2x6V cells, depending on perspective, but at the base level it's the 2V cells that are chemically independent of each other that I consider the "cell".
For instance, using the first "vat of chemicals on a bench", you cannot package 2 chemical vats together and maintain discrete properties, or make them portable and usable (other than wiring them together on the bench as separate entities, which makes them batteries of a sort, but since they aren't really usable in a portable sense, that doesn't fit batteries (might as well plug into the mains in that case).
Portable is a bit subjective, but if you can use a crane to load a 20kv 10kAh battery pack into a submarine without harming the batteries and no additional care needs to be taken in operation (other than preventing extreme stresses obviously) then it's portable... if you can't risk bumping the table or tilting the container without damaging the structure, it's not in my opinion portable, and therefore not a battery.