The standard cup mute can be written for in two configurations, the standard is the “loose cup” which players will default to if just instructed to use cup mute.
Loose cup will have the adjustable part of the cup mute further away from the bell than the “tight cup” which creates a more muted soft sound.
Most modern cup mutes have an adjustable cup but keep in mind that some older models do not allow you to adjust the cup.