In Pure Land Buddhism, what realm do the pure lands belong to? Are they outside of the usual six realms? Do the pure lands belong to heaven realms? Or a completely different realm from the normal six realms?
From wikipedia:

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.


However,


  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.