In the sentence presented, the word "conscientious" is used to describe the action of being and acting carefully or without forgetting anything as the sentence explains "she was careful, or conscientious" and the conjunction "or" shows equivalence between these two words, which means the word "conscientious" acts as a synonym of "careful". Therefore, this word most likely means "thorough" that refers to being attentive or careful. This idea is reinforced by "a few errors slipped by" that show despite this woman was conscientious she still has some errors.
