Firstly, I would say, the 3rd example is incorrect altogether.
The difference between “the” and “a” is similar to は and が, in that, there’s a particular feeling of emphasis that each word imparts to a sentence.
The air is sweet with the smell of spring.
This sentence, to me, feels more like your emphasis is on the end of the sentence. You want the listener to imagine the smell of spring.
The air is sweet with a smell of spring.
This sentence feels more like your emphasis is on the beginning of the sentence. You aren’t trying to make the listener imagine the smell of spring. You want them to know the air smells sweet, like if it were springtime.