Consider the following sentence from Oxford learners' dictionary, under their entry for the:
There was an accident here yesterday. A car hit a tree and the driver was killed.
In the first sentence, an occurrence of an accident is mentioned. Then, why "a car" is used in the second sentence? I think it should be "the car". Since it is assumed we are talking about of a car involved in the accident.
The dictionary doesn't say that the sentence would be wrong if the word the was used. In fact, I think all four of the following sentences are grammatical and sensible, although they each conjure up slightly different images in my mind:
A: There was an accident here yesterday. A car hit a tree and the driver was killed.
B: There was an accident here yesterday. The car hit a tree and the driver was killed.
C: There was an accident here yesterday. A car hit the tree and the driver was killed.
D: There was an accident here yesterday. The car hit the tree and the driver was killed.
Version A (the original) is correct because the first sentence never mentions a car. (The accident could have been a motorcycle accident, or a bicycle accident, or a van hitting a pedestrian, or even a small plane crash.)
Version B (your suggested version) is not incorrect, but it assumes the speaker already knows the accident was an automobile accident involving a single automobile. (This is not outside the realm of possibility; I can imagine, say, two police officers who were already made aware of the accident before they drove past the site.)
Version C (my new version) might be said if two people were standing on the sidewalk viewing the accident site. If there was only one tree nearby and they were both looking at it, the definite article could be used in front of tree, but we might be inclined to use an indefinite article in front of car if the car itself had already been towed away.
Version D (another new version) might be said on the scene, particularly if the car is still smashed against the tree.
If the first sentence was rephrased, though, you'd have a stronger argument for using the car instead of a car:
E: There was a car accident here yesterday. The car hit a tree and the driver was killed.
This would work especially well if there were several trees in the area and the speaker did not know which tree was hit.
an accident, a car
The accident on X Street or Road, the car or a car, depending.
The car is random in the first and could be specific in the second.
The Oxford's Learners Dictionary is 100% spot on, of course.
Compare:
Of the multiple bad accidents that occurred here yesterday, the car that hit a tree ended with a dead driver. Meaning that several accidents happened, but the specific car that hit the tree resulted in a fatality.
There were many bad accidents here yesterday, but the accident where the car hit the tree had a fatality. The larger point Lambie was trying to make, I think, is that we can use the definite article to single out one out of many. Similarly: There are eight eggs in the carton. The cracked one needs to be thrown away. We often use the definite article to call attention to one out of many. (A cracked one needs to be thrown away is also grammatical – but we can use the definite article there.)