We normally use 'light drizzle' instead of 'fine drizzle'.

For heavy drizzle, I'm more likely to say 'rain', but I hear 'heavy drizzle' used in weather forecasts so that's good.

You could say either, but fine drizzle would probably be used more.  Heavy drizzle, well that's just rain.  Drizzle signifies not raining very much, so to say heavy drizzle would perhaps be a bit odd, but entirely possible.

Another phrase is "raining in the wind".  I'd say it signifies less rain than a fine drizzle.