Class 0I uses a separate ground path. Example: semi-permanently installed equipment bonded with a ground spike.
Class II I'm not sure what counts as layers, possibly it's air inside the enclosure plus the enclosure thickness. A failure of the enclosure isn't obvious, would that be cracking or fragmenting or completely falling off? I'm not sure offhand how that's counted (and I'm sure it varies by standard).
Class III is SELV to SELV, so there's no need for insulation. An example product might be a DC-DC converter module, with functional insulation rated something low, like 100V. Such a device might be used where isolation is needed but high voltage is not, like breaking ground loops in communication systems, like Modbus (e.g., RS-485 crossing throughout a facility, so that it picks up far more than the +/-7V or whatever common mode range the receivers can handle).
Might also be used for PELV or TNV circuits, depending on rating. Ethernet for example, is transformer coupled, with, functional or basic type insulation, I'm not sure which; and a 1.5kV isolation rating. This covers ESD (which might be a 150pF 8kV source, which gets divided into a 1nF capacitor, reducing it to about 1kV peak); but does not very well cover induced lightning or mains transients (2.5kV+), so a long run (say, between facilities) or an accidentally cross-wired circuit (e.g. Ethernet cable gets nicked and touches mains) might not do so well.
Using such a device, on a circuit expected to have transients, may require protective devices to prevent their breakdown, and that then requires consideration of where the protection current flows.
For example, you can put MOVs or GDTs on a telephone line (which makes long runs between facilities -- expected to experience induced lightning surge), but the telephone set either needs to be grounded (so the fault current can be dissipated safely) or double insulated (so there's no risk to the user).
Regarding S/PELV and such: the CAPITALIZED TERMS are all described in the front section of these standards. Just look up the rest of the standard for definitions. Very boring reading, but eh.
SINGLE FAULT is one such term. A ground fault for example, is where a ground conductor has failed (broken). This might happen through injury or corrosion of a cable or connector. Obviously, if the device depends on sinking faults to ground, they will no longer be safe when this happens. This would violate the single-fault-tolerant property of SELV. The above hypothetical telephone, with a metal chassis and safety ground connection, would not be safe if it had just one ground wire.
Which, we have actual examples of these: the old fashioned telephone booth, mostly made of metal. I assume they are (semi-)permanently installed in such a way that grounding is redundant, therefore a single failure in any given connection does not lead to unsafe operation. (They may also be subject to standards much more stringent than e.g. IEC 60950-1 or the like; I have no idea.)
So, if you're just reading bits and pieces of standards, yeah it's not going to make much sense, go look up the whole standard if you can find it (or if you NEED it, don't be an idiot, BUY IT!).
Most of IEC has been adopted -- and made public -- by India. I don't know about EN standards. There may be compatible or similar standards in other systems that can be found.
Tim