God gave the 10 Commandments in Exodus 20:1-18. The people were so scared at the voice of God, that they asked God to stop talking to them, and asked Moses to go up and find out the rest of what God was going to say, and then tell them what it was. (Exodus 20:19) So, Moses went up and found out what the rest of the covenant was. This is the part of the covenant that is recorded in Exodus 21-23. You can confirm this by reading Exodus 24:3-8, to see that Moses read the "Book of the Covenant" (not just the 10 Commandments, but also the part they didn't want God to speak out to them), and they agreed to it. In the part that Moses read to them, we find this verse: Exodus 23:19 "“Bring the best of the firstfruits of your soil to the house of the Lord your God. “Do not cook a young goat in its mother’s milk."
Moses went back up to the mountain, where God was giving him the pattern for the Tabernacle (Exodus 25-31), the people got worried that Moses maybe something had happened to Moses. (Exodus 32). They made a false god, and held a festival to that false god. It also said that they were eating and drinking at this festival. (Exodus 32:2-6)
God told Moses to go back down and deal with situation. God was so angry with the people He was going to destroy them and start over with Moses' children as the children of the promise. Moses begged for the people's lives, and God relented. (Exodus 32:9-14)
Moses got so mad at the Children of Israel when he saw the party to the false god, that he threw down the tablets of the 10 Commandments that God gave him. He destroyed the false god, and killed the people who would not agree to recommit their lives to the true God. (Exodus 32:25-29). Then Moses went up the mountain to see if God would renew the covenant with the Children of Israel, after they had broken it so quickly.
Now, in Chapter 33 we see that God was willing to renew the covenant with them, if they would take off their jewlery. (Ex. 33:4-6)
So, we see the children of Israel had broken part of the original covenant, by making another god, instead of worshipping YHWH. They had used jewlery to make this false god. They had proclaimed a false festival to this false god. They had eaten and drunk and partied by this false god. We don't know what they ate, but we can surmise what it was they ate, by reading what God said, and emphasized out of the original covenant, when He decided to renew the vows.
In Exodus 34, God is renewing the vows, and asking the children of Israel who are left, who had broken the first covenant with him, to make a new commitment to Him. It makes sense that He would emphasize the laws that were the problem, and make sure they would agree to them. In Exodus 34, we see restating these laws: I will list the original source of the law in parentheses, to show where it was in Exodus 20-23.
1. Do not make molten gods. (Exodus 20:1-4, Exodus 34:17) They had made a molten god.
2. Keep the Feast of Unleavened Bread in the right time of year. This is the Festival that shows when and how I brought you out of Egypt. (Exodus 23:25, Exodus 34:18). They had said this false god they had made had brought them out of Egypt. (Exodus 32:4)
3. All the firstborn of the sheep and oxen belong to God. Redeem the donkeys and mules. Redeem your firstborn sons. (Exodus 22:30, Exodus 34:19. The rule for donkeys and mules was in Exodus 12:23, as a reminder that God brought them up out of Egypt.) They certainly didn't remember who brought them up out of Egypt, and we can surmise they may have sacrificed other animals other than sheep or oxen. We hope they did not sacrifice any of their sons to the false god.
4. You are to work six days, and rest on the seventh day. (Exodus 20:8, Exodus 34:21). This has to do with holidays that are set up by God. They set up their own holiday, and declared a rest day to their false god. They also may have not been resting on the day that YHWH set up, and needed to recommit to this.
5. And thou shalt observe the feast of weeks, of the firstfruits of wheat harvest, and the feast of ingathering at the year's end. (Exodus 23:16, Exodus 34:22) They had set up their own feast day, and God wanted them to re-commit to His ordained feast days.
6. Three times a year the men shall appear before God. (Exodus 23:17, Exodus 34:23). This has to do with the festivals God set up. They had set up their own festival to the false god.
7. Don't offer the sacrifces with leavened bread, and don't leave any of the Passover sacrifice until morning. (Exodus 23:18, Exodus 34:25). We can see that they had broken the specifics of the other commandments above. From this, we can surmise that the sacrifices they brought to their false god included leavened bread, or that they ate it on the next day. For whatever reason, God felt this statute needed to be part of the vow renewal.
8. The firstfruits of the land you are to bring to the house of God. (Exodus 23:19, Exodus 34:26) This is part of the firstfruits offering, and festival. They had set up their own festival to their own god.
9. Do not boil a kid goat in his mother's milk. (Exodus 23:19, Exodus 34:26). For the same reasoning as number 7, since we know they had broken the other points above, we can surmise that during their eating and drinking to this false god, they had broken this command of the original covenant, and it needed to be agreed to again.
I hope this helps show that the new set of Commandments was not a different set of the original Covenant. These commandments were part of the original set they had agreed to, and were the ones they had broken by making, and worshiping the golden calf. It also shows that God wants us to obey the entire covenant, listed in Exodus 20-23, and not just the part that is listed in Exodus 20.
Since Moses had broken the original set of the 10 Commandments, God wrote another set. All of these specific parts of the renewal of the vows can be included in the original 10:
1. Do not make other gods and worship them. (Exodus 20:3-4)
2. Remember the Sabbath day (Exodus 20:8), and His other sabbath days listed in the other festivals. Worship God in the way He wants to be worshiped.
Jesus said all the commandments can be summarized as "Love God, and Love each other." Matthew 22:37.
These all fall into the Love God category, in doing the things that He wants us to do.