Oh thanks for mentioning that! I have a RO unit I use for drinking water and was going to use it for Caridina shrimp, good to know that before I ordered them!
You mention [not] all have the remineralizing cartridge, how do I check/know if it does or doesn't? I suppose I can use the same test kits that I use for the fish water (API liquid test kits, pH, KH, GH, will get a TDS meter very soon)
Water tastes clean and drinkable to the taste buds, no nasty/funny taste to me. Maybe it does have the remineralizer or it doesn't and I just like the taste?
You have to ask or check, or if you see the physical unit the cartridges are labeled.
Your basic RO system consists of a pre-filter which removes sediment (it has no active chemical components), a carbon filter for removing various chemicals (including most chlorine), and then the RO membrane for removing most of the dissolved solids that remain.
Some systems add additional carbon filters before for more thorough use with chlorinated water (especially water with chloramines which are more widely used all the time).
This is all the basic "RO" part.
The ones that remineralize will then have a separate cartridge that is plumbed AFTER the RO membrane. So even if not labeled, seeing a cartridge after the membrane (assuming it is not a DI cartridge) is usually a remineralization cartridge.
On a distantly related note, home systems which have pressurized accumulation tanks are generally not as good for aquariums as otherwise, if you are trying for the most pure water. In any RO system, when the flow of water is stopped, there is a certain amount of seepage through the membrane (which depends on pressure differential to work -- when water stops flowing the pressure equalizes). Every time a RO filter is stopped for a bit of time and then restarted, the water for the first few minutes is much higher TDS than in normal run time.
If you are trying to make really pure water, it is useful to discard the first 2-3 minutes of run from the RO membrane when you start, and then don't stop it until you are done. If you have RODI filtration, it is actually helpful to have a divert between the RO and the DI part and let the first couple of minutes run off before switching it into the DI filter -- the DI is likely to remove all the excess stuff if you do not, but it will eat more into the lifetime of the DI media.
I guess I should also mention in all of this that for 99% of the people using any of these for fish or plants it just doesn't matter. Most tap water is just fine for most aquariums (suitably treated for Chlorine of course). It is always worth asking "what specific problem am I trying to solve", unless you (like me) are a bit OCD like me and just insist on trying for zero. Among the conditions that really need an RODI filter are very high PH (and you really need low PH, really need it), and tap water that varies a lot from day to day (and so you would be hitting the fish with lots of changes every water change). Most fish and almost all common plants deal just fine with high and low TDS, reasonably high and low ph, etc. They don't like rapid change, but can acclimate, even if their "native' land is very specific. Remember, most fish are dozens or hundreds of generations from their native lands and have already acclimated to some other environment, as many are raised in outdoor ponds that for darn sure don't use RODI water, and most of the rest in huge indoor facilities that also rarely use RODI water.
But if you decide you want to do RODI, it's nice to study it a bit; if you are going to put all the expense and effort into it, it is a shame to waste all that time and effort and get water that isn't really pure, even if you then just dirty it up again to put it in the tank (i.e. remineralization).