If using the word president in a sentence is it capital. Refering to our countries' presidents.

You only capitalize president when it is used as a proper noun, as in:

"Give this note to President Obama."
"Please respond to the question, Mr. President."

You don't capitalize it otherwise:
"Hilary Clinton is running for president."
"The president of the class is Timmy."
"One day even you can be president."

The rules given and the examples are all correct.

Yes, when referring to a specific country's presidents, the word "president" should be capitalized. This is done to give proper emphasis to the position and to show respect for the office.