What is the term “God” meant to mean in the DC universe?
Exactly what it means in this universe...someone/thing/power who is worshipped as a "God".
The only difference is that in the DC Universe (and Marvel Universe for that matter) is that some "Gods" are real.
What they are is unclear or rarely explored or just waved away. In the MCU for instance, "Gods" such as Asgardians are treated as more advanced aliens who were revered and worshipped as gods by more primitive man.
The issue of actual "divinity" is essentially ignored.
In the DCU this has not really been explored yet but at the moment, the Athenian gods referenced appear to be beings of great power and abilities. Whether they have any divinity is a matter of interpretation and faith.
The reference to "False God" in BvS was to highlight that the common folk were starting to worship Superman as a 'saviour' type character even though he would be the first to state that he is not a god.
So, just having enormous power (super or otherwise) does not make you a god but it might get you worshipped as one.