The monsoon is a singular word that describes all the storms that form during the summer. Monsoon is derived from the Arabic word "mausim" which means "season" or "wind-shift". The wind shift refers to the seasonal change in overall wind direction.
In Arizona winds shift from the northwesterly cool, dry winds of winter to the southerly winds of summer that draw in moisture off the Pacific, Gulf of California, and Gulf of Mexico. This moisture is needed for monsoon downpours.
Since the definition of the word monsoon is season the correct reference for the collective summer storms is simply the 'monsoon'. It is not incorrect to say season, it is just redundant. The individual thunderstorms are called monsoon thunderstorms or monsoonal thunderstorms.
The Tucson National Weather Service records monsoon data from June 15th to September 30th. However, the heavy downpours in Tucson generally hold off until the beginning of July. That is when the deeper tropical moisture gets into place over Southeast Arizona. This is why dew points are important to track when the monsoon begins.
As a general rule a monsoon dew point of 54° or higher in Tucson means the deep tropical moisture has moved into the area and there is a good chance of heavy rain if the atmosphere is favorable for storm development. The higher the dew point, the better chances of seeing strong storms.