No I quoted the actual decision. I don't know what you are talking about with "their version." Here is the link:
http://law2.umkc.edu/faculty/projects/ftrials/conlaw/loving.html
Yes the laws were racist, but the decision cites the 14th amendment 15 times and equal protection eight times. This decision was absolutely 14th amendment related.
The last line of part 2 of the decision says: "
Under our Constitution, the freedom to marry, or not marry, a person of another race resides with the individual and cannot be infringed by the State."
They are saying that the state cannot pass an interracial marriage ban without a compelling legal reason. The state didn't have one and the law was unconstitutional.