Law of God vs. Law of Christ
Answer: The first, "Law of God," is often used in the N/T to denote the Law of Moses. This, in contrast to the "Law of Christ" which is the Gospel — the Spirit of Life.
Many are understandably confused by the idea of "the Law." Often, it is used to describe a system of mere faith alone or "faith only."
In Paul's Letter to the Romans, the apostle demonstrates that there certainly are rules, regulations, commandments — obligations we must follow for salvation "in Christ." These are very much what we would consider the "Law of God."
However, Christianity is also generally referred to as the "Law of Christ" as per the OP (1 Cor 9:21 & Gal 6:2). This is elsewhere described as the "Law of Liberty" (Jas. 1:25, 2:12), and while this new law frees us from the Law of Moses, this does not mean that we are exempt from a much higher standard in Christ. The Lord's brother, James, describes this for us:
James 1:25: "But one who looks intently at the perfect law, the law of liberty, and abides by it, not having become a forgetful hearer but an effectual doer, this man will be blessed in what he does." (emphasis added).
James tells us that we must be "effectual doers" in this perfect law. The vital obligations we read in the New Testament cannot simply be ignored. While the OP suggests that the phrase "Law of Christ" only appears twice in the New Testament, as observed, it is also characterized as the "Law of Liberty" or the "Law of the Spirit."
The apostle Paul also spoke of those who were "without law" — that is, without the Law of Moses, yet they were still under the N/T Covenant of Christ as we see in the next two sets of passages:
Romans 8:2-4: "For the law of the Spirit of life in Christ Jesus has set you free from the law of sin and of death [the Law of Moses]. For what the Law could not do, weak as it was through the flesh, God did: sending His own Son in the likeness of sinful flesh and as an offering for sin..." (emphasis added).
1 Corinthians 9:19, 21: "For though I am free from all men, I have made myself a slave to all, so that I may win more... 21to those who are without law, as without law, though not being without the law of God but under the law of Christ, so that I might win those who are without law."
What is the distinction Paul is making above? Clearly, being "without the law of God" means being without the Law of Moses. This is differentiated from the "Law of Christ," or this passage would not appear as it does. Freedom in Christ means freedom from the Law of Moses, although that hardly exempts us from our new, higher accountability in Christ. Everyone on earth is subservient to Christ. Here is another designation for the Law of Christ:
James 2:8: "If, however, you are fulfilling the royal law according to the Scripture, “YOU SHALL LOVE YOUR NEIGHBOR AS YOURSELF,” you are doing well."
James assists us with another characterization: "the Royal Law." We should understand that the Gospel is this new, royal law. And, we must obey it under threat of eternal punishment. Hear the words of the apostle Peter:
1 Peter 4:17: "For it is time for judgment to begin with the household of God; and if it begins with us first, what will be the outcome for those who do not obey the gospel of God?" (cf. 2 Thessalonians 1:8-9).
What was that? It was yet another designation for the Law of Christ: the "Gospel of God." Obeying God is obeying His Gospel: the N/T. And, we need to understand that all of the commandments of Jesus Christ are to be adhered to:
John 14:15: “If you love Me, you will keep My commandments."
Let us consider an example of this:
Luke 22:19: "And when He had taken some bread and given thanks, He broke it and gave it to them, saying, 'This is My body which is given for you; do this in remembrance of Me'" (emphasis added).
This is the Lord's Supper. It is communion. Do you think the commandment by Christ — God in the flesh — to "do this" was merely a suggestion? Hardly.
In case one believes that the commandments of Christ are not binding, or have no legal status, they should think again. Read the words of Christ Himself:
Matthew 5:17-19: “Do not think that I came to abolish the Law or the Prophets; I did not come to abolish but to fulfill. For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished. Whoever then annuls one of the least of these commandments, and teaches others to do the same, shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven" (emphasis added).
Let us pause to observe 3 points about this set of passages:
I. Early in His ministry, Christ says that "not the slightest letter or stoke shall pass from the Law until all is accomplished." What exactly, was to be accomplished? Is this not His Own Sacrifice for humanity? YES, Christ came to "fulfill": he would fulfill the Law of Moses.
Upon His resurrection, the new Law of Liberty was instituted, the Law of Moses being nailed to the Cross (Col. 2:14). The distinction between fulfilling and destroying is analogous to the difference between paying a note of debt and abandoning it: in both cases the debt is effectively removed.
However, Christ took the Law of Moses out of the way. He did so not by violating it but by completing it! This was accomplished through 1) His perfect obedience, 2) His fulfillment of His role as the promised Messiah, and 3) His elevating all the O/T laws to a higher, purer standard: The Law of Christ.
II. Christ tells us that those who violate one of His commandments, has violated them all. His "new commandments" would comprise the Gospel, that was subsequently written after His Resurrection and later Ascension to heaven.
III. Note that in verse 19 of Matthew 5, Christ tells us that anyone who: "[does and teaches]" the commandments of Christ, is he who "shall be called great in the kingdom of heaven." There are phrases used against those who try to follow Christ's guidelines: "He is a legalist." This is an epithet charged against those who reject the false teaching of salvation through "faith alone" or "faith only," and strictly adhere to Christ's Doctrine.
Two laws are suggested by the OP. And, indeed, there are two: 1) the Law of Moses, that from which we have been liberated "in Christ"; then there is the Gospel; the Law of the Spirit; the Royal Law; the Law of Liberty, or simply: 2) the Law of Christ.
There is a gaping difference between these two laws. The first is often referred to as "the law of sin and death" — which is a misnomer because that law is neither sinful nor "deathful". Rather the terms refer to the effect of its holiness as being "sin's occasion" and "death's warrant."
Some may note that on the first day that Moses brought the Law down to the people, 3,000 were put to death:
Exodus 32:28: "So the sons of Levi did as Moses instructed [to kill those out of control], and about three thousand men of the people fell that day."
This is in stark contrast to the first day that the Spirit of Life in Christ went into effect. On that day 3,000 people 1) heard the word, 2) believed in Christ, 3) repented of their sins, and were baptized "for the remission of sins" to become saints (Acts 2:37-47).
Three thousand died at the inauguration of the Law of Moses; 3,000 "came to life" upon the unveiling of the Law of Christ. A great distinction indeed.