Do the pure lands belong to heaven realms? Or a completely different realm from the normal six realms?
About Pure Land of Pure Land Buddhism:
Upon entry into the Pure Land, the practitioner is instructed by Amitābha Buddha and numerous bodhisattvas until full and complete enlightenment is reached. This person then has the choice of returning at any time as a bodhisattva to any of the six realms of existence in order to help all sentient beings in saṃsāra, or to stay the whole duration, reach Buddhahood, and subsequently deliver beings to the shore of liberation.
About the Sphere of No Form:
Bodhisattvas ... are never born in the Ārūpyadhātu even when they have attained the arūpadhyānas.
About the Pure Abodes tier of the Sphere of Form:
Because a Śuddhāvāsa deva will never be reborn outside the Śuddhāvāsa worlds, no Bodhisattva is ever born in these worlds, as a Bodhisattva must ultimately be reborn as a human being.
In the rest of the realms of the Sphere of Form the beings continuously dwell in the state of mind corresponding to one of the four dhyanas, and therefore arguably cannot possibly be "instructed by Amitabha Buddha and numerous bodhisattvas".
This leaves the Sphere of Desire (kamadhatu) of which the only appropriate place for Pure Lands is the world of gods (aka Heavens). Indeed, the famous Tusita heaven is where the Bodhisattva Natha currently resides, who will later be born as the next Buddha, Maitreya.
Tuṣita (Sanskrit) or Tusita (Pāli) is one of the six deva-worlds of the Kāmadhātu, located between the Yāma heaven and the Nirmāṇarati heaven. [...] In Mahayana Buddhist thought, the Tuṣita Heaven is where all Bodhisattvas destined to reach full enlightenment in their next life dwell for a time.
Tuṣita is part of the same world-system as Earth, and so is relatively close, whereas the Pure Land of Amitabha Buddha is treated as a separate world-system entirely.
From all of the above it follows that Pure Land of Pure Land Buddhism are inside "the usual six realms", namely they are part of the heaven realms.
There are so many confusion regarding Pure Land Buddhism. Many people even see it as non Buddhist because it seems to be about the Amitabha Buddha saving humans through the power of his grace.
I am from a non denominational Chinese Mahayana organization with Chan lineage but we also heavily devote to Amitabha and emphasize the Amitabha Sutra in our practice.
After more than a year of chanting the sutra, my conclusion is that it is a metaphor for the Dharma! Remember that Pure Land Buddhism is an outgrowth of Mahayana Buddhism and therefore one of the highest goals is the Perfection of Skillful Means (Upaya Paramita). Mahayana texts utilize fantastical imageries and stories to bring across the difficult to understand technical concepts and teachings in early Buddhism, preserved in the Pali Canon and the Sanskrit Agama.
Imagine your average person being told about the Four Noble Truths, Suffering, Cause of suffering, End of Suffering and the Noble Eightfold Path. It doesn't interest him - "Give up my desires? That's ridiculous! And end of suffering doesn't sound exciting at all! Impermanence sounds so depressing!"
So this is where Pure Land Buddhism comes in. "What if we tell you that there is a place called the World of Supreme Bliss better than any worldly pleasures? And to reach there you need chant this sutra everyday, and I promise you Amitabha Buddha will appear before you". If you didn't realize, chanting is actually a very powerful form of meditation, in fact for the novice, it is faster and more effective than sitting meditation and trying to watch your breath. Here your attention and focus is on the sutra, attaining Right Effort, Right Mindfulness and Right Concentration. With that it is possible to experiences the mental factors of the Jhana/Chan/Zen, joy, happiness, peace, and realizations. So in fact the Pure Land is a metaphor for the enlightenment factors.
Further more it is stated in sutra that in the Pure Land of Amitabha, even lower realms (hell, hungry ghost, and animals) do not exist, this is a Mahayana message that ultimately the goal is not only to save yourself, but purify the whole world. Pure Land Buddhists vow to turn this very world into a paradise. And have a Pure Land in their hearts wherever they maybe.