Bonjour!
Yeah, this is a bit of tricky one. I'm going to do my best to explain it robustly.
The rule states that partitive (du, de la, de l', des) and indefinite (un, une, des) articles are reduced to de in a negative construction. A base example as you noted is as follows:
“J'ai une voiture” becomes “je n’ai pas de voiture”.
A) Construction: A construction refers to a single construction or clause. This clause may be a full sentence in itself or one clause within a sentence:
1) J'ai une voiture. (This is one clause, one construction, one sentence)
2) Je ne crois pas qu’on ait des devoirs. (This is one sentence, but two clauses separated by qu')
Based on this definition, then des in example #2 could never be reduced to de because it is not part of a negative construction or clause, it is part of a positive, subjunctive clause. Je ne crois pas is a separate negative clause that is not in the subjunctive but rather the indicative. Thus the rule does not apply.
B) Number of verbs: The rule states that the reduction occurs in a negative construction (clause), therefore it is essentially irrelevant how many verbs there are the clause and thus which verb is directly being negated. For example:
1) Je ne vais pas lire de livres. Lire is not being directly negated, however the rule still stands.
2) Je ne vais pas voir d'ami. Voir is not being directly negated, however the rule stile stands.
That distance from the directly negated verb is elongated in our example sentence, however it remains a single negative construction:
Je n’ai pas le temps de réserver de table.
C) The difference between de and un/une: Un or une can be kept if we want to emphasise the concept of “not one single”. For example (in order of degree):
1) Je n'ai pas de voiture. (I don't have a car.)
2) Je n'ai pas une voiture. (I don't have a single car.) - more emphasis, greater degree
3) Je n'ai pas une seule voiture. (I don't have a single car (at all).) - even greater emphasis / degree
Une could be kept in our example sentence, however it would technically change the meaning of the sentence. For example:
1) Je n’ai pas le temps de réserver de table. (I don't have time to reserve a table.)
2) Je n’ai pas le temps de réserver une table. (I don't have time to reserve a single table.)
In the example used by RobertC106: je n'ai pas assez d'argent pour acheter une voiture. This technically means that I don't have enough money to buy one single car. But you could equally say, je n'ai pas assez d'argent pour acheter de voiture, which means I don't have enough mony to buy a car.
D) The difference between the two is simply stress and emphasis. I think this is largely where the confusion lies and I think there are two points to be made:
1) The further the article and object are from the directly negated verb in the construction, the more unclear the rule feels, even for native speakers.
2) You can actually use either de or un/une, however they change the degree and emphasis. Both options being grammatically correct introduces a level of confusion.
For these two reasons, you will probably hear both forms being used interchangeably in spoken French, as people don't pay much attention to the nuance. However, in written they should technically be clear.
That is quite a long winded reponse, but I hope I have managed to break it down and build it back up in a way that makes sense.
I hope this helps,
- Mitchell