It is amazing how this can be brought to quite a different discussion by picking out a single word and then setting up a whole view on it. There were three different masters quoted, all of them have extensive teachings to see how they understood (and still do) that the essence of Mahamudra and Dzogchen are no different.
"Some people make distinction between dzogchen and mahamudra, but this is unnecessary because, fundamentally, they are two names for the experience. Different lineages simply employ slightly different methods of instruction that lead to the same realization."
(Kalu Rinpoche: Luminious Mind, p. 221)
While the approaches to the three views of Madhyamika, Mahamudra and Dzogchen are slightly different, their end results are identical.
(Chokyi Nyima Rinpoche: The Bardo Guidebook, p. 79)
Both dzogchen and mahamudra are methods for meditating on the mind. ... So we see that there is little difference between mahamudra and dzogchen.
(Thrangu Rinpoche: Essentials of Mahamudra, p. 10-11)
The meditation instructions of Mahamudra and the Six Yogas of Naropa can be traced to Rangjung Dorje, the Third Karmapa. It is said that the Dzogchen instructions come from Jigme Lingpa and his nyingthig teachings. Where did Jigme Lingpa get his teachings? These were derived from the Longchen Nyingthig, which was written by Longchenpa. Where did Longchenpa get his instructions? He received his instructions from Rangjung Dorje. So we can say that Rangjung Dorje is the source for both the Mahamudra and Dzogchen teachings. In his two shorter branch texts (Distinguishing Consciousness from Wisdom and A treatise on Buddha Nature), Rangjung Dorje says that whether one does Mahamudra or Dzogchen practice, buddha nature is the foundation from which both of these meditations develop.
(Thrangu Rinpoche: On Buddha Essence, p. 1-2)
But whether you follow Dzogchen or Mahamudra, please understand that ultimately there is no real difference. There is not one awakened state called Mahamudra and a separate one known as the Great Perfection. It is all of one taste within the expanse of dharmakaya. What these two words actually refer to is the basic nature of all things. Since all phenomena, all that appears and exists within samsara and nirvana, have the stamp of great bliss, it is called "the Great Seal," which is the literal meaning of Mahamudra. Similarly, since all phenomena are perfected in the expanse of self-existing awareness, it is called Dzogchen, or Great Perfection.
(Adeu Rinpoche: Freedom in Bondage, p. 32)
There’s a very interesting joke that I would like to tell you. Two masters of Mahamudra and Dzogchen were having a debate. The Dzogchen master said, “The Dzogchen is much more profound. It’s the highest path and the highest yana. There is nothing above. If you practice in your sleep, you can attain enlightenment. If you practice while you are awake, you can attain enlightenment. Even if you don’t practice, you will attain enlightenment if you have the karmic connection. So Dzogchen is much higher.” Then the Mahamudra teacher said, “Well, that sounds very good, but the Mahamudra teachings talk about one instant. In one instant, if you recognize it, you attain enlightenment in that same instant. If you fail to recognize it, then you are confused. Therefore, Mahamudra pointing-out is much more powerful because you can attain enlightenment in one instant. You don’t have to go to sleep to practice, you don’t have to wake up to practice, and you don’t have to hang around to attain enlightenment.”
(Dzogchen Ponlop Rinpoche: Penetrating Wisdom, p. 49)
"There is no such thing as the real mind. Ridding yourself of delusion: that's the real mind."
(Sheng-yen: Getting the Buddha Mind, p 73)
"Neither cultivation nor seated meditation — this is the pure Chan of Tathagata."
(Mazu Daoyi, X1321p3b23; tr. Jinhua Jia)
“Don’t rashly seek the true Buddha;
True Buddha can’t be found.
Does marvelous nature and spirit
Need tempering or refinement?
Mind is this mind carefree;
This face, the face at birth."
(Nanyue Mingzan: Enjoying the Way, tr. Jeff Shore; T2076p461b24-26)