WITH ALL OF THIS NOW IN MIND I WILL NOW DO MY BEST TO ANSWER YOUR ORIGINAL QUESTIONS: the Eastern Church always partakes of both the Body and the Blood, but the Western (Roman Catholic) Church developed a slightly different insight into the matter. In both East and West, both the Body and Blood must be offered, but within the Roman Catholic Western Church the priest alone must consume both because he acts in the Person of Christ at certain parts of the Mass. The moment that best represents the sacrificial moment of Christ’s death is precisely the separate focusing on “this is my Body” and, later, “this is my Blood,” as mentioned earlier. When these are said, it is that exact moment, a re-entry and re-presentation of that moment wherein Christ died for us. Without this, according to the Roman Catholic Church, Christ’s Eucharistic Sacrifice does not take place (the Assyrian Church of the East, another separated ancient Church within the East, however, does not have “this is my Body” and “this is my Blood” but their Liturgy was still recognized as truly anamnestic by Pope John Paul II…this is clearly a special case). While the priest must offer the Sacrifice, acting in the Person of Christ, and eating therein, the lay people are not ordained priests. There is, therefore, a distinction that was made between “performing the Sacrifice” and receiving the “Sacrament” (it’s like two sides of the same coin, so to speak). The Sacrifice is the Sacrament, but certain people perform certain parts of it (priests both re-present the Sacrifice and receive the Sacrament, yet the lay people, or even other clergy present but not presiding over the Liturgy, receive just the Sacrament). Eucharistic Sacrifice + Eucharist Sacrament = Same Eucharist. Remember, the Roman Church makes distinctions in order to unite. Within the Sacramental aspect of the Eucharist, Jesus is fully present under either the Body or the Blood because we are eating the resurrected Christ (his Body and Blood, which were separated, are reunited in his resurrection). By eating just the Body, you also are drinking the Blood, even though you didn’t actually partake of the Chalice. The Church, however, allows priests to distribute both the Body and the Blood, should the priest desire to do so. Another argument in favor of distributing only the Body is that the priest risks spilling the Blood. I hope this makes sense.
All of this, so far, provides an answer to the questions raised by the website that you mentioned.
Now in regards to the Old Testament Law against consuming “the flesh WITH the blood” of an animal, it will be shown that Jesus, the same God present in the Old Testament, did not break his own law. The prohibition against eating the flesh with the blood was a penitential secondary law, added alongside of the True Law (the 10 Commandments and original Liturgical Laws) because of the golden calf idolatry episode. Jesus had to quarantine the people from even further from idolatry, so he instituted this law. But why? Many idolatrous rituals, which the Israelites were addicted to within Egypt, involved consuming blood with the flesh of the sacrificial animal because, even for the pagans too, “the life/soul/power was in the blood.” If you wanted, say, the fierceness of a lion in order to cause fear in your enemies…drink its blood, sacrificially (that’s my own made up example to illustrate the point). It was forbidden because idolatrous sacrificial worship and partaking of idolatrous sacrifices were a communion with demons, leading to possible demonic possessions in the life of their worshippers. Within the New Covenant, however, Jesus fulfils this distorted but anticipatory longing of the pagans, by making the Eucharist not just a Banquet shared with Jesus, but also Jesus’ own Blood, containing his Divine Life and Power. He wants us to receive these. Jesus did not break his law. He fulfilled it. Just to note, however, eating blood (in terms of pagan sacrifice) is also condemned in Acts 15, however, again, this is only in terms of idolatrous sacrifice. If you are, like my wife, a fan of blood pudding, go ahead and eat it! lol…there’s no idolatry therein since there’s no idolatrous sacrificing going on!