What are Aṃṡāvatāra and Āvēṡa-avatāra?

The Ramanuja school of Srivaishnavas has an interesting supposition about Vibhava-avatāras (incarnations or descent). Sri Sri Sri Tridandi Chinna Jeeyar Swamiji explains the “categorization” of avatāras of Sriman Narayana in his “Ramayana Pravachanam - Part 36” (~49:00 onwards). For example, amshavataras and aveshavatars. The following is a brief transcription from Telugu to English.

They are arranged in a, sort of, hierarchy.

Amshamsha avatara (aṃṡāṃṡa): A very minute part of a part. For example, the avatāras of great King Pṛthu Cakravarti and Ṛṣabhadēva. They are not God Himself, instead, individual souls who’ve been granted a minutest of minute ability (sāmarthya) of the Lord. They’ll do great things which previous men have not done, owing to some grace of God giving them special powers (ṡakti). Swamiji continues to narrate the doings of King Pṛthu. Like they say “na viṣṇuḥ pṛthivīpatiḥ” – without Viṣṇu’s mercy (kṛpā), one cannot become the lord of Earth (king). Such avataras are usually perfomed for a long time and specific purpose. [Similar is the case with Buddhāvatāra?].
Amshavatara (aṃṡāvatāra): They are God Himself, descended for a specific purpose, but short-lived. For example, killing of demon Hiraṇyākṣa in the case Lord Varāha, or gracing Prahlāda Mahārāja in the case of Lord Narasiṃha. The incarnation disappears once the purpose is served. We have no additional information as to what happened to that avatar before or after.
Avesha avatara (Āvēṡāvatāra): God does not appear directly, but instead takes posession of or “enters” or “encompasses” (āvēṡa) another individual soul (jīva). These avataras are usually long-lived, performing multiple purposes, but the āvēṡa itself is ephemeral (comes and goes). As long as the jīva is thus subsumed, he can perform great miracles. As soon as the possession is let gone, he cannot do it again. Like a magic ring on some people empowers them with special abilities. There are two types of Āvēṡāvatāra:
- Svarūpa āvēṡa: Lord Paraṡurāma was a person, a sage (ṛṣi) but he is not the Lord Himself. Lord Viṣṇu engulfed His own essential nature/character (svarūpa) in him. Because of this, he was able to circumnavigate the world 21 times and decimate adharmic kṣatriyas. During Rāmāvatara, Lord Viṣṇu decided that it is no longer necessary to continue Paraṡurāma’s āvēṡa, so He took back His svarūpa from Paraṡurāma when taking the ṡiva-dhanus from his hand. Paraṡurāma asks Lord Rāma to take back all his puṇyas. Similar is the case with Lord Balarāma, who is also svarūpāvesa avatāra but not a sākṣāt avatāra. When Lord’s avesha was taken back, during the end of avatara, Balarāma ended up with drinking wine and enjoying – because there was no āvēṡa of the Lord. Balarāma is none other than Ananta-ṡēsa, the serpant who occasionally had āvēṡas of Lord Viṣṇu come and go.
- Ṡakti āvēṡa: Instead of His own nature (svarūpa), the Lord injects His energy or potency (ṡakti) in other beings to get the job done. Arjuna is an example, that’s why he was able to fight wars and win. Likewise, Vedavyasa bhagavān was given the strength to classify the Vedas – it’s an ephemeral ṡakti (tātkālika). Otherwise he’s a normal maharshi.
Pūrṇāvatara: Meaning, a complete or accomplised avatara in which all the nava-rasas are enacted in His līlā. Lord Ṡrī Rāma is the only one such complete avatāra.
Paripūrṇa avatara: Meaning, a fully complete or perfected avatāra. The one which includes features, activities, līlas of all other avataras. Lord Ṡrī Kṛṣṇa is the only such perfect avatāra. For example, hāsya rasa is not frequently noticed in Rāma (Valimiki Ramayana records a couple instances); Rāma is sober and modest, but hāsya is more evident in Kṛṣṇāvatāra.

Though Swamiji didn’t say, it seems that not all avatāras are equally worshippable in mūrti form in Kali Yuga. I mean, for example, we don’t have a sampradaya/tradition of worshipping Pṛthu Cakravarti.

Again, though Swamiji didn’t say, all the avatāras are equal in principle (tattvataḥ); it’s just that He didn’t display all His power in all His avatāras – He chose only what was necessary as per His will.