Booth's is a summary of soldiers that fought for Louisiana during the CW. Not 100% correct, but pretty close. So it is interesting that it shows a different history. Then again, it could be a different Robert Smith. I also looked up an Edwin Smith, but nothing for this name in the 2nd La Cav in Booth's.
You can also order a copy of his military records too. A link can be found at the top of the page. This should settle the question.
JIM T