Capacitor discharge and 1/2 terminal voltage sources
The one-terminal source is referred to ground, which has a specific symbol that you also used.


The 1-terminal supply is like a battery in which the negative terminal is automatically connected to ground. The 2-terminal is more like a battery: you can stack more or put it inside a circuit, so it simply creates a voltage. Then you have to reference it to ground if you want.
In the first circuit, the capacitor is connected between the low terminal of the voltage source and ground, so basically it's not supplied by anything. You have to put a ground symbol in the negative terminal of the voltage source, to make the circuits equivalent.
As I said in n. 1, since the 2-terminals supply is not referenced, you are not creating any potential across the capacitor.
Especially in this case, yes, 0 V is ground.


P.S. It's not the best to put horizontal grounds, consider turning it.

This is the equivalent of your second circuit with a 2-terminal supply