I'm so confused by this rule I just read; its from The Elements of Style by Strunk & White.

The rule says:
6. Do not break sentences in two.

"I met them on a Cunard liner many years ago. Coming home from Liverpool to New York."
"She was an interesting talker. A woman who had traveled all over the world and lived in half a dozen countries."

Their explanation: In both these examples, the first period should be replaced by a comma and following word begun with a small letter.

What I don't understand is why these two sentences even need a comma? Isn't the second half of the first example a dependent clause? And isn't there a rule that says that if a dependent clause comes after the independent clause then a comma isn't needed? I thought the rules for using commas in sentences were:
1. Independent clause+Comma+conjunction+Comma+Independent clause
2. Dependent clause+comma+Independent clause
3. Place commas in between non-restrictive elements

Those are all the comma rules I know, so I have no idea where to place the rules that Strunk & White are talking about. If someone can clarify this for me it would be greatly appreciated.

The sentence below needs a comma because it is parenthesis in apposition.

"She was an interesting talker, a woman who had traveled all over the world and lived in half a dozen countries."

In other words, talker = woman.

The sentence below gets a comma under a spin-off of the non-restrictive ruling.

"I met them on a Cunard liner many years ago, coming home from Liverpool to New York."

The comma basically says "I am a clause that does not modify the thing to my immediate left."

These examples make it clearer:

I swallowed a fly riding my bike.
I swallowed a fly, riding my bike.

It's a really great question. This topic is not well covered in books.