Why is transconductance defined so?
The ‘trans’ bit is short for ‘transfer’, and the transfer referred to is from input to output.

So think of transconductance as a transfer function, which is always in the form \$\frac{output}{input}\$, and ‘conductance’ is \$\frac{current}{voltage}\$, hence \$g_m=\frac{output\:current}{input\:voltage}\$