You are correct; Sabbath and Saturday have the same meaning; seventh – that is the seventh day of the week. There are some groups who meet on Saturday, but most Christians meet on Sunday. We do not celebrate the Sabbath on Sunday however. We celebrate the resurrected Christ on the “Lord’s Day” – Sunday.
Jesus Christ was in the grave on the Sabbath – resting. He came out of the grave on Sunday – referred to in the New Testament as the Lord’s Day because it is the Day Christ arose. Without the resurrection “we are of all men the most miserable”. (1 Corinthians 15:19) In the book of Acts we see the New Testament church meeting on the first day of the week – the Lord’s Day.
I believe that the law was given to the Nation Israel in part to prove that mankind could not keep the law. See the books of Romans and Galatians – we are not under law, but under grace. Again, I believe that Christians are to live by a higher standard – one of grace. One example; In Matthew 5:27-28 Jesus said "You have heard that it was said to those of old, 'You shall not commit adultery.' But I say to you that whoever looks at a woman to lust for her has already committed adultery with her in his heart”. We have a higher standard.
Our calendar may or may not be correct, but to me it doesn’t matter because I’m not under the law because in Colossians 2:16 Paul said; “let no one judge you in food or in drink, or regarding a festival or a new moon or Sabbaths”.
I do hope this is of some help since I struggled with this issue years ago myself.