Lincoln's Signing of the 13th Amendment

On Facebook, someone criticized the Lincoln movie for inaccuracy on the ground that the President said he wanted to sign the 13th Amendment, but presidents don’t sign amendments.

But the movie is correct: Lincoln signed the 13th Amendment.  The interesting question is why?

I can imagine various reasons.  Here are three possibilities:  First, Lincoln didn’t want to take a chance that the courts would hold his signature was necessary and so signed it out of an abundance of caution; second, Lincoln wanted to affix his name to symbolize his personal connection to the Amendment; third, President Buchanan had signed the proposed Corwin Amendment in 1861 (which would have protected slavery), and so Lincoln wanted to mirror that action for the amendment abolishing slavery.

But these are just speculations.  As far as I know, this was the only time a President signed an amendment that became enacted.  Does anyone know why he did so?