Are there different types of enlightenment?
The are the 3 Bodhis: Sammasambuddha, Pacceka-Buddha, Arahath. Then there is Sopadisesa-nibbana and Anupadisesa-nibbana. But I don't think that's what you are asking about when you say different types.

What you need to keep in mind is that same thing can be described in many different ways. To give an analogy, an elephant can be described as 'heavy' by some, 'tall' by some, 'large' by some, 'gracious' by some, 'noisy' by some etc.

One person can say "spreading the light" to which another might say "dispelling the darkness". But they both refer to the same thing.

On the other hand, different teachers and traditions might label different experiences as enlightenment depending on their understanding or misunderstanding. That does not mean there are many types of enlightenments. Otherwise the Buddha would have listed them down under the Third Noble Truth.


  enlightenment as a shift in identity


This is cutting off of identity-view(Sakkayaditti) when one attains Sotapanna. It is also partial liberation from delusion.  


  liberation from greed, hatred and delusion;


This is when one attains Arahanthship. So it appears that Jack Kornfield might have gotten confused by the descriptions of different stages of enlightenment and different stages of mental purifications.


  devotion and surrender, being carried by the Buddha’s “grace";


This is not a description of enlightenment in Theravada Buddhism. But it may just mean that if one devoted oneself to the Buddha and submit to the practice of holly life under him, one will attain enlightenment.


  There are many other gates: the gates of compassion, of purity, of
  surrender, of love. There is also what is called the “gateless gate.”


There are only 3 real gates. Others may bring one close to enlightenment, but the meditator still has to attain absolute certainty of either impermanence, suffering or non-self to get to Nibbana. Read here for more details.


  This is enlightenment as love.


This maybe referring to attaining the Jhanas through Metta meditation. But there is more work to be done from there to reach enlightenment.


  This is enlightenment as emptiness


This maybe talking about ākiñcaññāyatana(infinite nothingness), the 7th Jhana or the 3rd Arupajhana. But it's still not enlightenment. 


  He expresses enlightenment as fullness.


This maybe talking about the rapture and joy arising in the first jhana or the fruits of Buddhanussati meditation: "When a bhikkhu is devoted to this recollection of the Buddha, he is respectful and deferential towards the Master. He attains fullness of faith, mindfulness, understanding and merit. He has much happiness and gladness." -Visuddhimagga


  Enlightenment as mindfulness.


Mindfulness leads to enlightenment and Arahaths are said to be always mindful. But mindfulness itself is not enlightenment.


  The Dalai Lama personifies enlightenment as compassionate blessing.


Dalai Lama is not Theravada, but he could just mean that enlightened beings are compassionate and a blessing to the world.


  He expressed enlightenment as wisdom.


Wisdom leads to enlightenment. Seeing impermanence, suffering and non-self is wisdom.