Leviticus 17:10-14 is where the specifics about eating blood is given. Verse 13 indicates eating blood is about not draining the blood from the carcass. I don't think it means eating rare meat. But in any case, this is old covenant which we are not under.
We have to read these passages within the context of the Acts of the Apostles. After Christ's return to Heaven, there were two opposing ideologies within the nascent Christian community. The first ideology of St. Peter purported that Christians should remain Jewish. They were to have a closer identity with the rules and regulations of the Old Law. The second ideology of St. Paul denounced the regulations of the Old Law as profiting one nothing. The passages in Acts 15 and Acts 16 appear to be instances of the first ideology of St. Peter, which disappeared quite early within Christian culture as Christianity became a hellenic religion rather than a strictly Jewish religion.