I want to get rid of 'article' confusion once and for all!

I know there are innumerable questions about 'articles' on this and other sites on the Internet. I always try to learn them but some or the other way, they play tricks on me. In other words, I am never positive when I use them, in short. If two articles weren't enough to spin my head, the concept of 'zero article' or 'anarthrous' served as another nail in the coffin! :)

At times, I understand them perfectly but sometimes, I commit blunders! I want to solve this once and for all. I thought a lot and then am finally coming up with some sentences that would help clear my (and many others'?) doubts (hopefully).

Help me understand the subtlety of these sentences.

1. A crown cork of a bottle


2. A crown cork of the bottle


3. The crown cork of a bottle



4. The crown cork of the bottle

My understanding about those sentences:

1. Any crown cork of any bottle (but then is that crown related to that bottle is the question!)


2. I'm going to a shop with a bottle in my hand and asking for a crown cork that fits the only bottle in my hand


3. There is only one crown cork in front of me. I'm telling that this is the crown cork that can fit any bottle


4. The bottle and the crown cork both are right in front of me. That crown cork fits to only that bottle.

Answerers, please mind that I haven't introduced any bottle or crown cork previously! I'm ready for detailed answers. :)

Answer

A less formal use of "a crown cork of a bottle" - bottles are presumed to have only one mouth:

1. We were playing in the back-yard when Jim noticed something shiny, a roundish sliver of metal half-dug in the ground. "It could be a crown cork of a bottle ," I said.

The non-standard use fixed by Jim to the more formal the + a combination:

3. "I see," said Jim. "But bottles have only one crown cork, so you'd better say the crown cork of a bottle."

The "standard" use of the definite article in relation to things already introduced (the bottle) or implied (the crown cork):

4. "My dad once bought a very fancy bottle of beer, and after opening it he gave the crown of the bottle to me. It had a nice emblem on it," I said.

A sentence I constructed to make use of "a cork of the bottle":

2. "Oh, yeah," said he. "I wish that bottle had two mouths. Then you could get a crown cork of the bottle, and I could get a crown cork too.

According to this Google Ngram proposed by Damkerng, your options 1 and 2 are either non-existent or rare as hen's teeth. But I'm not totally sure. If I'm wrong, let someone correct me.