It really depends on context. The specific Greek is

ὁ δὲ εἶπε "ὦ Κροῖσε, ἐπιστάμενόν με τὸ θεῖον πᾶν ἐὸν φθονερόν τε καὶ ταραχῶδες ἐπειρωτᾷς ἀνθρωπηίων πρηγμάτων πέρι." [Your translation is pretty literal]

I bolded "τὸ θεῖον" which is literally "the god" which in this case is generic (as you guess, no specific god is being referenced). But in Plato's Apology, for example, when Socrates refers to "o θεῖος" he is specifically referring to Apollo. It is really just a contextual thing.