So I did an internet search about power supply classes. I found a TDK document named:
"What is the difference between Class I and Class II power supplies?"
Quite at the beginning it says:
"Class I power supplies have an earth-ground connection, whereas a Class II product does not. "
Now looking at your block diagram the answer to your question is quite obvious. It took less than half a minute to find out.
The presence of a PG terminal means nothing, it has to bonded to accessible parts & circuits to achieve class I protection. Presumed SMPS output is SELV circuit, SMPS must be class II isolated.
As we all know, That schem in top post has earth to pri and sec side. Incidentally The best way to do that, from an EMC point of view, is to connect a sufficiently insulated wire across the isolation barrier, from the incoming earth wire of the mains plug, and connect it to the secondary side.
The wire should be rated to carry the fault current prior to fusing. (and obviously insulated)
Usually, if it requires an earth/ground connection, you would presume, for safety reasons, it is class 1. The trouble is with many cheap supplies, presumption could be dangerous... I would also take any labeling, such as CE marks to be dubious.
Check with the manufacturer, as previously stated and use supplies from decent suppliers...
Safety first, test the supply...