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Basically you should always use "14th Dalai Lama" even though the Dalai Lama is that same continuous Awakened being incarnating as Dalai Lama, because each and every Dalai Lama has a different ego, body, and often temperament and purpose in life. Also to answer your main question, they also never live at the same time i.e. 13th has to die before can incarnate as 14th.
If you are mentioning the "14th Dalai Lama" solely in your writing, then it is OK to shorten it to just "the Dalai Lama." Even better than this is to avoid the convention altogether and refer to him by his real name "Tenzin Gyatso."
This naming convention also goes for the countless Buddhas. The Buddha of our planet is "Shakyamuni" (the jewel of the shakya clan") or referrable by other more specific personal names.
There's some good books and articles out there showing each Dalai Lama's life. It makes one think... "when I become a bodhisattva, what incarnations will I take on to complete various karmas and purposes?"
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It should be "14th Dalai Lama" or "14th Dalai Lama", not "14 Dalai Lama".
I think that "Dalai Lama" is a title, not a name.
According to Wikipedia,
The 14th Dalai Lama (religious name: Tenzin Gyatso, shortened from Jetsun Jamphel Ngawang Lobsang Yeshe Tenzin Gyatso, born Lhamo Dondrub,[1] 6 July 1935) is the current Dalai Lama
Another example of a title is "President of the United States". For example, Barack Obama is "the President", and he is also more specifically called "the 44th President".