Note that Moses specifically cites the words spoken to the assembly by God himself as what is written on the tablets. Both Exodus and Deuteronomy record God speaking the 10 Commandments directly to the people
Deut 5:1 And Moses summoned all Israel and said to them, “Hear, O Israel, the statutes and the rules that I speak in your hearing today, and you shall learn them and be careful to do them. 2 The LORD our God made a covenant with us in Horeb. 3 Not with our fathers did the LORD make this covenant, but with us, who are all of us here alive today. 4 The LORD spoke with you face to face at the mountain, out of the midst of the fire, 5 while I stood between the LORD and you at that time, to declare to you the word of the LORD. For you were afraid because of the fire, and you did not go up into the mountain. He said:
vs 2-21..THE TEN COMMANDMENTS... ending with
Deut 5:22 These words the LORD spoke to all your assembly at the mountain out of the midst of the fire, the cloud, and the thick darkness, with a loud voice; and he added no more. And he wrote them on two tablets of stone and gave them to me.