Another other JPS source on this question refers to the first ten commandments in the original post as the universal decalogue that applies to all people, Jews and non-Jews, and is symbolized by its utterance in the wilderness away from national borders.
In Exodus 34, the second list of ‘ten commandments’ in the original post, are referred to as the ‘cultic decalogue’ those commands which distinguish the Jewish religion from others.
Again, they are all part of the 613 commandments found in the Torah (as I understand it, as Judaism teaches it) but this is not even close to being a fatal blow to the credibility of scripture, certainly no “discrepancies” here, when one commits oneself to the study.
Certainly the approach to scripture study by both Judaism and Christianity involves faith, patience, humility, perseverance, harmonization, awe, etc. to love God with one’s whole heart, soul, mind, and strength.