Removing a charge counter from Sigil of Distinction is the only way to pay for its equip ability. If it has no charge counters on it, its equip ability can't be activated.

If Sigil of Distinction is attached to a creature when its equip ability is activated targeting a different creature, the first creature will immediately get a little smaller because a charge counter has been removed from Sigil of Distinction as a cost. Then, when the equip ability resolves, Sigil of Distinction will move to the new creature and the first creature will lose the entire bonus the Equipment was giving it.