Class I, Class II were original intended just for mains, with Class III intended for devices powered by an unspecified, non-mains external supply at low voltages which are already safe, i.e. from a separate power supply that has it's own safety insulation which could be Class I or Class II.
For example, your mp3 player is a Class III device that can be charged from any 5V USB. If the USB supply itself has a mains part, it can be assumed to be safe based on regulations that normally apply to any mains connected power supply. That supply could be Class I (e.g. desktop PC) or Class II (small wall plug in device). Hence it is not meaningful to refer to the mp3 player as either Class I or Class II.
IEC 60601-1 mangled this as it so often does. They argued that Class III is not acceptable for medical devices, which is OK (mostly), but it should have created a new Class e.g. Class IIIS = powered by a specified separate supply.
Instead they just deleted Class III, which left it unclear as to whether secondary powered devices should be Class I or Class II.
In practice this is irrelevant. All the clauses in the standard that mention Class I or Class II are only relevant to mains parts. Sometimes this is not immediately clear and you need to check the definitions, or titles or preliminary comments. For example, in Clause 7.2.6 there is a requirement that Class II devices are marked with the double box symbol. However, the clause title is "Connection to the SUPPLY MAINS". 8.11.5 (fuses in the supply) similarly has a reference to "mains" in the clause title.
At it's core, "protective earth" itself is defined as a connection to the "external protective earthing system" (i.e. the wall supply). Class I and Class II are intended to indicate if this connection is important for safety.