I don't think that there is a clear division between bioreactor and fermenter and the terms are often used interchangeably, but they do carry a difference in connotation (and marketing). The term "bioreactor" implies complexity, research driven, and/or maximal cell growth. Bioreactors are more state of the art and are used primarily for studying microorganisms or producing difficult to grow cells or for peak production of cells- while the term "fermenter" implies production of a substance besides or other than just the cells and might have the connotation of utility. Fermenters also carry a historical connotation of brewing beer, wine, and other fermented food stuffs or materials. Also, it's important to bear in mind that bioreactors are used for fermenting, because bioreacting is not a real word.
If you just need a simple design for producing the bacteria or a bacterial product, then the term fermenter might be more appropriate. If you need computer control, or if you are using it to investigate/research growth and/or production parameters then you could call such vessels bioreactors without the risk of looking foolish. I think the best way to describe the difference is probably by example, but instead, I tried to itemize some of the distinguishing details, as follows:
Fermentors commonly have all or some the following elements: Agitator (of simple design), pH meter, temperature control (jacketed vessels are common), pipes for transferring nutrients, water, etc., and a sampling port (maybe). It may be made out of wood or clay, but if it is made from steel, plastic, glass, or other modern materials, that certainly wouldn't disqualify the term fermenter, but I would probably not call an oak barrel or a clay pot a bioreactor.
Bioreactors generally have the same things as fermenters, but bioreactors also usually have some or all of the following additional elements: oxygen probe, air or oxygen pump(s) and diffuser(s), other gas ports, additional ports/pipes for pH monitoring and control (often a programmable computer controlled pump is used for multiple solutions), electrolyte balance, radial or axial impellors- which operate at hundreds or thousand of RPMs, a vapor condensor (maybe), spectophotometer (for cell density measurements), nutrient processing data, and any variety of data measurement and/or recording is the often the point of interest, or when growth of tricky or finicky cells is necessary (especially like hybridomas, phytoplankton, or perhaps generally eukaryotic cell culturing). Also, there are different kinds of bioreactors like wave bioreactors (often in bioreactor bags on shaker tables instead of having an impellor).
In general, sterility is more important for bioreactors whereas many fermenters (or fermentation products, like yoghurt) can kind of take care of themselves, or a certain "cleanliness" will work. In both cases, steam sterilization is common. Again, the division is a bit blurry, and the terms might be used interchangeably by some people, like a jacket (light) versus a coat (heavy).
So, for your specific example, it would definitely be a fermentation system (not a bioreaction system), but regarding whether to call the vessels bioreactors or fermenters is more of an opinion, I'm afraid; but personally, I would probably say the fermentation system has two fermenters, since the idea is to produce a product and it sounds like they may be simple vessels. However, if the system has on-the-fly computational checks and balances, then I might be tempted to call the vessels bioreactors (if dazzle demands it).
Also, the smaller (200 L) vessel could be called an inoculation vessel without any connotations.