In the vaidika tradition it doesn’t take the same color as in later tradition. The great emperor trasadasyu is termed ardhadeva but only that. While deva & martya are distinct, the deva can assume any form for particular reasons.
Thus indra assumed the form of AkUpAra the gigantic tortoise. indra also appeared to medhAtithi kANva as a ram. indra took the form of the hyaena to conquer the asura-s. indra took the form of a gautama v1 to spy on the asura-s. viShNu took the form of a vAmana to conquer the dAnava-s. It also seems that it was understood that the boar that bore the Earth was viShNu. The gods might appear as birds to sit on the barhiSh to take the offerings.
Thus, the daivI vibhava-s were clearly understood. This has deeper Indo-European roots: northern Germanics: Odin becomes a snake & eagle. Iranics: verethraghna assume 10 vibhava-s some overlapping with indra &viShNu. Thus, the original IE avatAravAda was what may be termed vibhavavAda: the gods taken on many distinct forms (vibhava-s) to accomplish one or another end.
I believe this was the original intent in the rAmAyaNa but it was presented in such a manner that the audience had to recognize the gods rather than the author plainly stating it: indra assumed a human form as rAma & viShNu as lakShmaNa. The other gods as the vAnara-s. This is finally “revealed” in mandodarI’s lament where she voices her suspicion that indra as taken on a human form to kill rAvaNa (keeping with the different vibhava-s he might assume in the shruti to achieve different goals).
Now the more modern avatAra-vAda came with the sAttvata branch of pa~ncharAtra & the pAshupata shaiva-s. The former were the first to more directly consider their heroes (& originally also the pANDu-s to a degree) as incarnations of the vyUha gods of their system. The incarnational principle is made explicit in their early text the bhagavadgIta. This idea grew in its importance & greatly influenced both various “core” Hindu streams & counter-religious streams.
The shaiva version might be termed AveshavAda – where lakulIsha somasharman is reaminated into life evidently by the entry into his corpse of the god rudra himself. A element of this preserved in the later shaiva siddha traditions including matsyendra.
To conclude traditions can be divided into:
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- Indo-European vaibhava tradition: quite a deep part of the religion;
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- the synthetic avatAra-vAda important for vaiShNava-s & lay H.
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- AveshavAda important for pAshupata-s
Thus, while quite distinct in expression some variant of this genre of ideas is fairly important to central in different H streams & counter-religions.
Sister traditions
It is a worthy topic – there are many dimensions to it in my own analysis:
- the avatAra as the divine entity in a “material” body (dominant in H);
- A variation on this the son of the god – Herakles, pANDu-s, vAnara-s.
- descent of the abstracted divinity Xvarena – Iranic.
- The demi-god – ardhadeva – like trasadasyu who presumably became one via acts of great valor – the material being intersecting with the daiva by act that go way beyond the ordinary individual. A debased version might have been seen in cults of certain Greco-Roman figures.
Outside the IE realm, such a concept might be seen in shinto & many folk expressions in India. One could also include counter-religious figures like the tathAgata and the nagna in this category.