Yes, Precipitation is one of the most important criteria but not the "only" criteria. As mentioned above, there are many others criteria.

A quick look on criteria by IMD is here-

http://www.imd.gov.in/section/nhac/dynamic/MOK.htm

A few papers by Prof. PV Joseph (and associated crossed references) can be helpful 

http://iri.columbia.edu/~ousmane/print/Onset/Josephetal94_JClim.pdf

http://iri.columbia.edu/~ousmane/print/Onset/Josephetal06_IntJClim.pdf

Since monsoon onset is usually defined as a specific day within a month, your data of monthly resolution would not be able to resolved this. Perhaps consider using daily or atleast pentad data. 

Monthly data is not good to define the monsoon onset date. The daily or pentad data meet this purpose better. 

Also, monsoon onset has various definition, like by zonal wind, by rainfall, by olr, by relative humidity ...

But, please do not get confused by som many index. They are describing the same process so that they should be consistent with each other. Which one is the best? Oh, think about the story about blind people touching elephant ...

You may have interest in reading my papers discussing monsoon onset over Bay of Bengal, which represents the earliest monsoon onset in Asian region. Visit my ResearchGate page.

As suggested above, monthly rainfall are not adapted to define monsoon onset. It depends also on what you want to analyze. Are you interested by crop impact (date of sowing ?) or by a large-scale definition of onset for meteorological purpose ? For the first aspect, you need daily rainfall data but for the second one, you can define the onset from reanalyses or equivalent product.

I agree with the idea that daily rainfall data are much appropriate than monthly rainfall data for estimating the monsoon onset and subsequent monsoon delay (we are at intraseasonal timescales!). You certainly know that the monsoon onset may also be estimated from other indices than those based only on precipitation (OLR, wind,  wind direction ...) If a can give you any advise, you can use different free  available products on the web:

--satellite  daily rainfall (then average over a large area by bands of laitude)

- satellite outgoing longwave radiation (then average over a large area by bands of latitude)

Then you construct an index (for instance excess of precipitation north of a latitude compared to precipitation south of this latitude; the same for OLR). You can find some ideas and good reference in the following paper on the West African monsoon:

Caniaux, G., H. Giordani, J.L. Redelsperger, F. Guichard, E. Key, and

M. Wade, 2011: Coupling between the Atlantic Cold Tongue and the West

African monsoon in boreal spring and summer. { J. Geophys. Res..,

116, C04003, doi:10.1029/2010JC006570.

Regards

G. Canaiux

The monsoon onset and monsoon onset delay must include the interaction of the moisture with the Pakistan-Arabia Dust Cloud.  I am putting together a new group to start talking about those interactions at https://www.researchgate.net/post/New_group_to_discuss_monsoon_plus_Dust_Cloud_and_Pseudomonas_interactions?_tpcectx=profile_highlights