I believe that people refer to $\alpha$ as the exponential map exactly because of the "big example (I)" in Pottharst's notes. Also, a small point, for any scheme we can give many log structures. For instance, in addition to the main example of a log str associated to a closed immersion, the simplest examples are $M = \mathcal O_X^*$ and $M = \mathcal O_X.$ I found K. Kato's paper "Logarithmic structures of Fontaine-Illusie" to be a good introduction.

Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press, Baltimore, MD, 1989.