Again, One of your examples under the heading -
la moitié de [quelque chose] = half -the- [something]
was “Il ne mange qu'une moitié du biscuit.”
Can you clarify, then, why response “Tu as bu une moitié de la bouteille.” was marked incorrect and the suggested correct response was “la moitié de” ?
I am afraid there is no grammatical explanation for this, it is stylistic. The French just tend to say "la moitié" more often than "une moitié".
According to the lesson, "la moitié de" means "half the [something]", and "une moitié de" means "one half of [something]". I'm not sure if this is also true in French, but in English I think you can only say "one half" if it is possible to separate the thing into two halves. A biscuit, a cake or an apple, say, can be cut or split into two, and you can eat one half and leave the other. But you can't do that with a bottle. You can eat one half of the biscuit (a KitKat for example), but you can't drink one half of the bottle.
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