We demonstrate that the timescale of charge removal by lightning is an essential parameter for the criterion of sprites along with the charge moment. Sprites can be initiated under small charge moment changes of several hundreds of Coulomb kilometers if the discharge time is sufficiently short as hundreds of microseconds, whereas they cannot be done even with 1000 C km if the discharge time is in the order of milliseconds. We generalize the criterion as an analytical expression of charge moment and discharge time and discuss its sensitivity to the ambient plasma conductivity based on the result of the classical electrostatic model simulation. Our result can explain almost the critical values of charge moment for sprite initiation recently derived from analysis of lightning-induced electromagnetic pulses. We deduce from our criterion that charge moment changes necessary for halo are smaller than those for streamers because the former can be initiated without significant ionization. Considering the role of charge moment and discharge time, we discuss further the condition for initiation of various classes such as a halo and a cluster of streamers with or without halo.