"I ate half of the camembert!"

Translation: J'ai mangé la moitié du camembert !

[Elon_the_Hittite]

I put 'une moitié', which was accepted (and somewhat surprised me). I reasoned that 'une moitié' would better represent a non-specific half, rather than 'la moitié', which would suggest a designated half.

[MikeJarvis4]

How should I think about the distinction between demi and moitié?
I thought demi also meant half but there must be some distinction that's important because it didn't accept it.