Personally, I frown upon the game usage of "avatars" for deities, that is, the idea that deities are fundamentally intangible, but run various puppet-versions of themselves throughout the world(s). This idea is basically at odds with the majority of real-world pantheistic religious beliefs and mythologies; for the Greeks, Norse, Egyptians, Sumerians, Celts, and so forth, the gods were particular non-divisible individuals. There are rarely stories or myths which display the gods using puppet-selves to present their image.
In fact, the idea of "avatars" is really a peculiar notion specific to one single religious tradition: that of the Vedic/Hindu. As James Ward and Robert Kuntz wrote in the original Deities & Demigods book for the Indian Mythos (1st printing, p. 75), "Probably the most difficult concept this mythos presents, at least in AD&D terms, is that of the 'avatar'. An avatar is a physical manifestation of a deity upon the Prime Material Plane..." It's ironic that since the time of that writing, the "difficult concept" for AD&D has become a presumption for all D&D mythologies of all sources.
Of course, the main reason this was done (around the advent of 2nd Edition AD&D) was to put the gods in a realm beyond injury by player characters. As James Ward wrote eight years later in the 2nd Edition Greyhawk Adventures book (p. 4), "Destroying an avatar does not harm a deity in any way". This may make some observers of the game more comfortable, but it is frankly at odds with most sources of mythology. As powerful as they might be, most gods (especially Western ones) were physical individuals who might be occasionally injured by mortals of sufficient potential and hubris. Achilles overcame a river-god by the name of Scamander before he battled Hector -- to say nothing of Heracles who lived as a man but at turns contended as equals with Helios, Apollo, Poseidon, and Ares. In a rage, Gilgamesh once injured the great goddess Ishtar by throwing a donkey's jawbone at her, forcing her to flee the scene. For the Celts, too, it has been written that "there are no hard and fast rules between gods and mortals; mortals can wound gods" (A dictionary of Irish mythology, p. 136). In short: using "avatars" to place the gods beyond the power of any mortals is a violation of the principles of most fictional and real-word pantheistic mythologies.