I think there may be a few different topics that are getting confused here. Many regulations and standards commonly classify the things they’re regulating into different groups or “classes,” so simply stating that something needs to be “Class II” is not sufficient to communicate requirements without also mentioning which standard or regulation is being referenced.
The information referenced above discusses one approach for protecting a patient against accidental shock or contact with a live electrical circuit. Issues relating to which wire types are considered suitable for use inside a device is a separate question, as Dave mentions. “UL wire styles” might be a useful search term.
Additionally, devices intended for medical use are subject to regulations regarding leakage currents. While insulation systems may prevent direct contact with a power source, there is always some degree of capacitive coupling across an insulation barrier through which AC current can flow and potentially deliver a harmful shock to a patient. A “medical” power supply typically will have a lower level of leakage current than non-medical types.