But "procession" is quite vague, and isn't the same as "avatar" or "incarnation." Everything in the universe would proceed from God in monotheism. The Holy Spirit in Judaism is just God's influence over everything, like the Logos in Stoicism.

Nothing was fixed in stone in the earliest churches. The theology of the Holy Spirit evolved up to the Nicaean Creed. Either way, the Holy Spirit isn't the same as an incarnation, a divine person in a human body. In comparing Christianity to Hinduism, I'm talking about divine incarnations, which are evidently the same as avatars.