As when learning any language, past a certain point, it's best to abandon grammar rules and focus instead on the information the writer or speaker is trying to convey.
Let's take a related example:
Her face momentarily flashed just the hint of a smile.
Her face momentarily flashed just a hint of a smile.
In the larger context there is likely no difference in meaning between these. The woman smiled for a moment. That's all. We move on with the story.
Nevertheless there is always a difference in nuance between using the definite and the indefinite article. You know the definite article is used to talk about things that are already known or mentioned, or things that are unique. "The hint of a smile" relies on the second usage. It (very slightly) suggests that this wasn't just some ordinary hint of a smile, but represents an archetypal hint of a smile.
Don't put too much weight on it, though. It's a subtle and mostly stylistic nuance that, again, does not change the meaning in any significant way. Let me give another example that does make a difference:
When talking of Laura, the young men would wistfully remark that hers was the face that launched a thousand ships.
"A face that launched a thousand ships" refers to the legendary Helen of Troy, who was so beautiful that, after she ran off with Paris, a prince of Troy, the Greeks launched an army to bring her back.
(Note: The actual story is more complicated than this, but a full discussion is not relevant here. If you aren't familiar with the "Iliad", I recommend the BBC series "Troy: Fall of a City", as it sticks fairly close to the source material).
Here the definite article is used because Helen's face is unique and well-known, and the men are saying Laura's face is just like Helen's. If instead we use the indefinite article, we'd have to change the sentence:
When talking of Laura, the young men would wistfully remark that hers was a face that would have launched a thousand ships.
which is to say, Laura's face is an example of a face like Helen's in that, if she had been alive in those days, she too would have caused an army to chase after her. The definite article points to a known example that actually had some effect, while the indefinite article makes it a hypothetical with the same presumed effect.