There are four stages leading to attainment or Nirvana:-
- Sothapanna - Stream-enterer
- Sakadagami - Once-returner
- Anagami - Non-returner
In each stage, different elements of the ten sanyojana (fetters) will be dealt with. The Pali canon's Sutta Pitaka identifies ten fetters as:
- belief in a self (Pali: sakkāya-diṭṭhi)
- doubt or uncertainty, especially about the Buddha's awakeness and nine supermundane consciousnesses (vicikicchā)
- attachment to rites and rituals (sīlabbata-parāmāsa)
- sensual desire (kāmacchanda)
- ill will (vyāpāda)
- lust for material existence, lust for material rebirth (rūparāga)
- lust for immaterial existence, lust for rebirth in a formless realm (arūparāga)
- conceit (māna)
- restlessness (uddhacca)
- ignorance (avijjā)
-AN10.13 - Saṁyojanasutta
Fetters, 1-5 are known as lower fetters where-as fetters 6 - 10 are regarded as higher fetters. Lower fetters gets removed progressively in Sovan, Sakadagami and Anagami states, while the higher fetters are removed in an Arhath.
All who attain Nirvana can attain it via either of 3 ways in terms of effort required to attain Nirvana:
- Savaka bodhi or Sravaka
- Pacceka bodhi or Pachcheka buddha
- Samma Sambodhi or Samma Sambudhdha
Sravaka bodhi requires the least effort to Nirvana, and thought of as the easiest and quickest path. Sravaka bodhi can only exist when there is Dhamma taught by Samma Sambudhdha is present in the world, as Sravaka (those who listen and learn Dhamma) can only exist when there is Dhamma present.
A Pachcheka buddha discovers enlightenment for himself, with own effort. A Pachcheka Buddha lacks the power to serve others by teaching Dhamma which he himself has discovered. Pachcheka Buddhas arise only during those periods when the teaching of a Samma Sambuddha does not exist. Pachcheka buddha spends more time than Shravaka in realizing Nirvana as it is done with own effort.
A Samma Sambuddha discovers enlightenment for himself, but in addition, he is able to teach Dhamma to others and show the path to Nirvana. Out of the 3 Bodhi, Samma Sambudhdha spends the most time. There are 3 categories of Samma Sambudhdha, in terms of the time required to attain Nirvana:
- Praknknadhika bodhisathwa
- Sadhdhadhika bodhisathwa
- Wiiryadhika bodhisathwa
In the above list each bodhisathwa spends progressively more time in preparation to becoming budhdha. Least time is spent by Praknknadhika bodhisathwa, where most time is spent by Wiiryadhika bodhisathwa.
All three (Savaka bodhi, Pachcheka buddha, Samma Sambudhdha) are equal in terms of enlightenment, i.e. their liberation from sansara, which means that these are not three different classes of Nirvana.
However, the Sravaka and the Pachcheka bodhi are regarded as inferior to the Samma Sambuddha in terms of other, special qualities and capabilities. The arrival of Samma Sambudhdha is regarded as the highest form of mangala (luck) to all beings, because Samma Sambudhdha opens the path to Nirvana for many Sravaka.
The decision is left to the individual whether to take the way of the Sravaka, Pachcheka buddha or Samma Sambuddha.
EDIT: After the edit from the OP with more clarifications.
In Theravada, only the Arhath is enlightened. Sovan, Sakadagami and Anagami states can be thought of as intermediate states leading to Nirvana. I think there is nothing wrong in thinking of them as levels of progressive understanding. However, there is no way for the practitioner to know which state they are in. So, there is no such thing as realization of 'I'm already enlightened' for Sovan, Sakadagami or Anagami disciple. It is believed only Lord Budhdha can evaluate someone's state of mind correctly and declare the state (Sovan, Sakadagami or Anagami) they are in. However, one knows when they are enlightened, until their journey ends in Nirvana, disciples must strive towards Nirvana.
If by 'intrinsic enlightenment' you mean seeing through the world we have built around a 'self', yes it is Nirvana according to the Theravada Buddhism.
Apologies to OP, I am not qualified to compare Theravada with hongaku theory. I hope this answer will help you perform the comparison yourself.