We need to look at the word truth a little bit philosophically:
When someone tells you the truth, there is only one, single possible version that is true. Not various possible truths. That’s why we don’t use the word “a” with the word truth. We use the word “the” because there is only one definite story that is the truth.
Lies, on the other hand, can be varied and many. That’s why you can say “He told me a lie”.
Half-truth is not truth! [color=blue]Those are two different words! A half-truth is ultimately not really truthful, but rather designed to deceive, is incomplete, and therefore it’s not that much different from a lie. There are various different ways you could tell a half-truth. People who tell half-truths are often just as creative as liars. There could be any number of half-truths connnected with a situation. So, you should treat half-truth the same way you treat lie i.e., use the word “a”.
When you say half the truth, then you’re back to simply using the word truth again. You’re only describing what portion of the whole truth was told.
Half the truth is just a shorter way of saying “half of the truth”. When the word truth is used, we’re speaking about a specific story. And so saying half the truth is like saying “half of the story”. (Half of a specific/definite story.)
Does that help?