The undefined command is
\square. This command is provided by the
amssymb package, which is loaded by
texmate. You can also load it yourself:
chessboard itself (i.e., Ulrike Fischer) made the decision not to load
amssymb, as explained in the changelog in the manual:
2011-03-17 (Version 1.7) Changed definition of the triangle mover
style. It now uses tikz and no longer amssymb. chessboard no longer
loads amssymb (it clashes with xunicode).
So apparently there could be some incompatibilities - but the MWE above works ok and produces the same output as your first picture.