But that is not correct. Both times the Lord wrote what he had spoken to them out of the midst of the fire. The scriptures even say it was the same writing.
Deuteronomy 10:4 And he wrote on the tables, according to the first writing, the ten commandments, which the Lord spake unto you in the mount out of the midst of the fire in the day of the assembly: and the Lord gave them unto me.
And I had not realized that the phrase "the ten commandments" was literally God's description of the words he spake to them. It was what the Lord spake to Israel as recorded in Exodus 20, and also recounted in Deuteronomy 5.