Are there different names given to volcanoes to describe their level of activity? If so, what are they?

Yes

Normal “Volcano is in typical background, noneruptive state
or, after a change from a higher level,
volcanic activity has ceased and volcano has returned to noneruptive background state.”

Advisory “Volcano is exhibiting signs of elevated unrest above known background level
or, after a change from a higher level,
volcanic activity has decreased significantly but continues to be closely monitored for possible renewed increase.”

Watch “Volcano is exhibiting heightened or escalating unrest with increased potential of eruption, timeframe uncertain,
OR
eruption is underway but poses limited hazards.”

Warning “Hazardous eruption is imminent, underway, or suspected.”

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