It is a multiplication factor, derived by dividing available average radiation at the Earth surface by averaged available radiation at the top of the atmosphere.
Yes, that is the equation HOMER uses for the average clearness index because that is the definition of the average clearness index. The averaging can be performed over any time period including one minute, one hour, one day, one week, one month, or one year. The twelve monthly values are averaged over one month, and the annual value that HOMER shows is averaged over one year.
The value that HOMER identifies as the annual average clearness index is just that, and we will call it nothing else. You are right that the annual average clearness index shown in HOMER is a weighted average of the twelve monthly values. It is not the simple arithmetic average of the twelve monthly values, and I will attempt to clarify that in the help file.
But note that the twelve monthly values of clearness index are also weighted averages. Imagine a place where every day in March is perfectly cloudless (clearness index 0.8) during the day and completely overcast (clearness index 0.1) during the night. The monthly average clearness index for March for that location will be exactly 0.8. It does not matter what the clearness index is overnight because no radiation strikes the top of the atmosphere at night. It is perfectly analogous to say that it does not matter much what the clearness index is during the winter because not much radiation strikes the top of the atmosphere during the winter.