Is the word piano short for a longer word?

Originally, yes. The predecessor of the piano was the pianoforte or fortepiano, a keyboard instrument designed for both control and sustain (like the clavichord) and variable loudness (like the harpsichord). The modern piano uses similar but not identical features.

Piano in Italian means 'quiet' or 'soft', and 'pianoforte' means soft/loud, a description of the keyboard instrument that was capable of varying the loudness of its sounds.