Questions about the countability and articles of "stench"

1. Countable vs. Uncountable

Cambridge Dictionary states "stench" is both countable and uncountable. CambridgeDictionary

When do I treat it as countable versus uncountable?

2. Articles (A/An vs. The)

Oxford Dictionary shows these two examples. Oxford Dictionary:

• an overpowering stench of rotting fish
• the stench of treachery

How do I choose between "a/an" and "the" in front of this word?

How do I choose between "a/an" and "the" in front of this word?

The same way you choose it in front of any word.

It seems like you recognise that the choice of article is directly dependent on whether the noun is countable or uncountable/unique.

"There is a man in your garden." (countable)
"It's the man who cuts my grass." (unique)

"There is a smell in here." (countable)
"It's the smell of bread." (unique)

Use the indefinite article when it is one of many or unidentified; use the definite article when it is something or someone specific.

• "An overpowering stench of fish" acknowledges that the smell is not unique. It has to, because if there is an overpowering kind, there must be other kinds.
• "The stench of fish" focuses on what the smell is - that it is unique, recognisably fish.