The original mantra is indeed "Hare Rama Hare Rama Rama Rama Hare Hare Hare Krishna Hare Krishna Krishna Krishna Hare Hare".
The scriptural basis of this mantra comes from the Kali-Saṇṭāraṇa Upanishad where Brahma reveals this mantra to Narada in response to the latter's question of how Jivas can resist the effects of Kali in Kali-Yuga:
At the end of Dvāpara yuga, Nārada went to Brahma and addressed him thus: "O Lord, how shall I, roaming over the earth, be able to cross Kali?" To which Brahma thus replied: "Well asked. Hearken to that which all Śrutis (the Vedas) keep secret and hidden, through which one may cross the saṃsāra (mundane existence) of Kali. He shakes off (the evil effects of) Kali through the mere uttering of the name of the Lord Nārāyaṇa, who is the primeval Puruṣa." Again Nārada asked Brahma: "What is the name?" To which Hiraṇyagarbha (Brahma) replied thus: (the words are:) "Hare Rāma Hare, Rāma, Rāma, Rāma, Hare, Hare; Hare Kṛṣṇa, Hare, Kṛṣṇa, Kṛṣṇa, Kṛṣṇa Hare, Hare." These sixteen names (words) are destructive of the evil effects of Kali. No better means than this is to be seen in all the Vedas. These (sixteen names) destroy the āvaraṇa (or the centripetal force which produces the sense of individuality) of jīva surrounded by the sixteen kalās (rays). Then like the sphere of the sun which shines fully after the clouds (screening it) disperse, Parabrahman (alone) shines.". (Kali-Saṇṭāraṇa Upanishad)
So yes, your family and the saints you met are right about the original mantra beginning from "Hare Rama". It was only modified by Gaudiya Vaishnavas because of their preference/importance to Krishna.
As a sidenote, I would like to add that in the mantra, "Rama" refers to the Maryada Purushottam Rama but according to the Gaudiya Vaishnava interpretation, "Rama" refers to Balarama, the elder brother of Krishna.