Some historical standards formed due to instrumentation. Cloud cover observations were setup based upon okta, which are eighth fractions of the sky covered overhead during the past hour (perhaps 8 observations were taken/recorded by early instruments/observers). This lead to the United States National Weather Service using this table for reporting and forecasting:
Likewise, the tipping bucket rain gauge measured in 0.01 inch (0.25 mm) increments, so that influenced how precipitation categories were chosen. This journal article states that:
In weather observations, drizzle is classified as (a) “very light”,
comprised of scattered drops that do not completely wet an
exposed surface, regardless of duration; (b) “light,” the rate of
fall being from a trace to 0.25 mm per hour: (c) “moderate,”
the rate of fall being 0.25-0.50 mm per hour:(d) “heavy” the rate of
fall being more than 0.5 mm per hour. When the precipitation
equals or exceeds 1mm per hour, all or part of the precipitation is
usually rain; however, true drizzle falling as heavily as 1.25
mm per hour has been observed. By convention, drizzle drops are
0.5mm or less in diameter”
Showing that there is plenty of connection in the definitions of rain/drizzle intensities to the 0.25 mm baseline.
However, while many organizations may work from this framework, there's no reason other organizations must base their wording and icons on these historical criteria. They may wish to have more or fewer categories in their forecasts. For example, Wunderground's API table shows that:
They use just four cloud coverage icon categories, while their forecast wording options include clear, cloudy, mostly cloudy, mostly sunny, partly cloudy, partly sunny, and scattered clouds. So they don't appear to match the standard NWS table.
So long story short: There are some historical standards, but sites can take liberties in reporting it how they wish. Looking through some of the APIs, it would definitely be nice if they would better indicate the actual criteria they use!