Yes, it does have a wide range of meaning. As pertains to the discussion, I am using it only to define those religious practices or beliefs which included blood sacrifice, actually or symbolically. The short list would include Mithraism, Judaism, and Christianity.
That's a unique usage of the word. I'm really not sure how it's a useful term anymore if you're going to redefine it in that way, so that Christianity (the thing the term "paganism" was invented to contrast) is included as a form of paganism. It's a bit like saying that a "reptile" is any animal with a face, including mammals.
I'm suggesting that Judaic animal sacrifice and Christian symbolic sacrifice are more formalized, 'authentic', and 'official' than their pagan counterparts.
If you're in fact suggesting that Judaic animal sacrifice was more formalized than pagan animal sacrifice, then I must say that you're mistaken. Greco-Roman religion is the pagan tradition that we have the most evidence for by far, and you're not going to find a more formalized example of sacrificial cult. It may have differed from Judaic sacrifice in a few particulars, but not to the degree that the average modern person would notice the difference. But one doesn't have to know much about it to know that, so I must be misunderstanding you here.
In particular, the ancients seemed to have held that the blood was the life-force; that it carried with it some kind of magical, transformative power without which life was non-existent. As I recall, some ancient warriors, upon slaying a legendary opponent, would drink his blood to gain his power. Animal blood had the power of transferring guilt and sin away from the sinner in Judaism, while Jesus is the scapegoat and 'Lamb of God' whose blood has the power to 'take one's sins away'. The idea of the scapegoat, found in Judaism, goes back much further to Syria in the 24th century BC.
24th century BC? That must be a typo, as we have no evidence for specific religious beliefs or practices going back that far, nor had goats even been domesticated yet.
But aside from that, we don't need to speculate about why ancient people sacrificed animals: they
tell us repeatedly, if you bother to read ancient literature. The gods enjoy the scent of the burning fat, which was often spiced with fragrant herbs, and the act of sharing a meal with humans gave them pleasure, as did the prestige that came with having people call on them for patronage. In most cases the animals would be slaughtered quickly, then bled, then butchered and cooked, then eaten by the celebrants. In most of those cases the blood was not used for anything, nor was it seen as particularly important to the process. In Judaic practice in particular, religious taboos against the consumption of blood meant that it had to be removed as much as possible prior to the cooking. In a few Greco-Roman sacrificial rites the blood might be used for something, but it's not consistent, and it's not clear that it's the blood itself that's important and not just the fact that it's been consecrated to a divinity.
As for the lamb metaphor, you're getting the basic sacrifice to Yahweh (normally an ox, at least in the 2nd Temple period) confused with the Passover myth. The lamb's blood was painted on doors to keep death from visiting the houses during the final plague in Egypt (in a noteworthy exception to the general rule about blood's being unclean). The Christ is compared to the lamb in the sense that he saves people from death. Sin doesn't enter into that equation directly, at least not as formulated by the early Christians who wrote the scriptures.
As I mentioned, the idea that Christ's death was a blood sacrifice to atone for sin is medieval in origin, and it shows in the degree to which its formulator didn't really understand the nature or purpose of sacrificial cult.
The bottom line of my argument is that there is no difference in the
significance between the blood sacrifice of Jesus and that of any other religious practice, as both imply a magical and transformative power of the blood.
*White magic has traditionally referred to the use of supernatural powers or magic for good and selfless purposes. With respect to the philosophy of left-hand path and right-hand path, white magic is the benevolent counterpart of malicious black magic. Because of its ties to traditional pagan nature worship, white magic is often also referred to as "natural magic".
White magic - Wikipedia, the free encyclopedia
It's a cute theory, but from an academic standpoint it rests on a general lack of understanding of how people actually saw these things in ancient times, and there's no real effort to engage with the piles of primary-source evidence we do have on the subject. Modern occultism, including the theory of magic that you cite here, is a recent invention that has no direct connection to the attitudes and practices of people in antiquity, and trying to interpret their beliefs and practices in those terms isn't terribly helpful.