I don't think it follows that, because it is not class II, it therefore is class I.Is it Class II? No, because it has an earth connection, and so will not bear the Class II construction mark. It's definitely Class I.
It does not meet the definition of class I either.
The fact that it has an earth connection is not the criterion.
The criterion is what is done with exposed conductive parts.
This is the whole point of John's original question - How are such items classed? - which no one seems able to answer.