Dtyjlui

The Hare Krishna mahamantra which I have heard from my family and saints is the following:

Hare Rama Hare Rama Rama Rama Hare Hare Hare Krishna Hare Krishna Krishna Krishna Hare Hare

Wikipedia mentions the same mantra and so does this source

While the one which I have heard from ISKCON is little bit different:

Hare Krishna Hare Krishna Krishna Krishna Hare Hare Hare Rama Hare Rama Rama Rama Hare Hare Hare

Although it doesn't make much a difference but still I want to know which is the correct one?

Some people say the latter one is interpolated. Since Krishna was more revered in the heart of some devotees, they interpolated it. Is that true?

Please note that the question is not whether interpolating this mantra slightly makes a difference or not but only that whether it has been done or not.

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.

Quoting from the book "Mahamantrakalpataru" by Sri Sitaramdas Omkarnath and also Sitaram-Leela-Alekhya by Promodranjan Gupta:

The original mantra given in the Kalisantaranopanishad started with Hare Rama and the second line started with Hare Krishna. It was a Veda mantra and meant for the Brahmins alone :"Sarvadaa suchirasuchhirvaa pathan BRAHAMANA salokataam sameepataam saayujyataam eti" meaning a brahmin chanting it in pure and impure state can attain different types of Mukti(Ref: Sitaram-Leela-Alekhya, By Prof.Promodranjan Gupta,Mahamilan Math,page 220). Mahaprabhu Sri Chaitanya changed the order making it stating with Hare Krishna so that any one can chant it, while keeping the result of chanting ie getting rid of the evils of Kali-Yuga and so being illumined unharmed. His version was supported by Radhatantra, Brahmanda-Purana andYogasara-Tantra.

So only for the Brahmins who obey all the scriptural orders, the version starting from Hare Rama is chantable.

Actually both the mantras, starting with Hare Krishna and Hare Rama are mentioned in our scriptures, but the mantra starting with Hare Krishna has been considered of highest importance because Krishna is the supreme personality of godhead according to the scriptures. Also Rama in the mantra is non another than Lord Sri Rama who appeared in treta yuga . It is also mentioned that Rama is another name of Sri Krishna which means one who gives pleasure to others and that's why he appeared as Lord Sri Rama in treta yuga. So It doesn't matter much whether you chant Hare Krishna first or Hare Rama first, but if you are considering the highest of all the mantras then you should chant Hare Krishna first.