In your question, it is not the number of times something is mentioned, rather

A

is used when being nonspecific, whereas

The

is used when being specific or pointing out uniqueness or utmost

I'm going to tell you about a city called Tripoli.
The city (that I just mentioned) has lots of beautiful houses.

"the city" is used since you are specifying the city you just mentioned in the previous sentence

This is a different pattern to

I'm going to tell you about a city called Tripoli.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.
It is the most famous and largest city in Lybia.

Where there are many cities with "beautiful houses" and "architecture", but only one is the "*largest" and "most famous".

The ordering could be changed to

I'm going to tell you about a city called Tripoli.
It is the most famous and largest city in Lybia.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.

A native speaker would not have a problem with either ordering.

Maybe this makes it clearer the difference in using "the" vs "a", but it is not dependent on the "ordering".

In your question, it is not the number of times something is mentioned, rather

A

is used when being nonspecific, whereas

The

is used when being specific or pointing out uniqueness or utmost

I'm going to tell you about a city called Tripoli.
The city (that I just mentioned) has lots of beautiful houses.

"the city" is used since you are specifying the city you just mentioned in the previous sentence

This is a different pattern to

I'm going to tell you about a city called Tripoli.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.
It is the most famous and largest city in Lebanon.

Where there are many cities with "beautiful houses" and "architecture", but only one is the "*largest" and "most famous".

The ordering could be changed to

I'm going to tell you about a city called Tripoli.
It is the most famous and largest city in Lebanon.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.

A native speaker would not have a problem with either ordering.

Maybe this makes it clearer the difference in using "the" vs "a", but it is not dependent on the "ordering".

In your question, it is not the number of times something is mentioned, rather

A

is used when being nonspecific, whereas

The

is used when being specific or pointing out uniqueness or utmost

I'm going to tell you about a city called Tripoli.
I'm going to tell you about the city called Tripoli.

both can be used without loss of meaning, the second has greater emphasis on a particular city.

Since many cities have "lots of beautiful houses", you can use

It is a city with lots of beautiful houses.
It is one (of many) cities with lots of beautiful houses.

whereas

It is the city to view pre-Ottoman architecture.
It is the best city to see pre-Ottoman architecture

It may be possible that your rule assumes a pattern where the first mention is general and subsequent mentions are more specific.

In your question, it is not the number of times something is mentioned, rather

A

is used when being nonspecific, whereas

The

is used when being specific or pointing out uniqueness or utmost

I'm going to tell you about a city called Tripoli.
The city (that I just mentioned) has lots of beautiful houses.

"the city" is used since you are specifying the city you just mentioned in the previous sentence

This is a different pattern to

I'm going to tell you about a city called Tripoli.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.
It is the most famous and largest city in Lybia.

Where there are many cities with "beautiful houses" and "architecture", but only one is the "*largest" and "most famous".

The ordering could be changed to

I'm going to tell you about a city called Tripoli.
It is the most famous and largest city in Lybia.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.

A native speaker would not have a problem with either ordering.

Maybe this makes it clearer the difference in using "the" vs "a", but it is not dependent on the "ordering".