Here's one example from a hydrology point of view: https://en.wikipedia.org/wiki/Water_year
And the UK Met Office has a nice definition and determination for different parts of the world:https://www.metoffice.gov.uk/weather/learn-about/weather/types-of-weather/rain/rainy-seasons
But you need to be careful because definitions may not be precise and just because there is a definition doesn't mean it can't rain or not rain at any time.
And in a time of climate change these things might be changing depending on location.