RG, if there is a relationship between visibility and rainfall rate it probably involves droplet size, e.g. when does light rain or drizzle transition to cloud/fog. This data is unlikely to be available for normal operations.
I cannot find any relationship between rainfall rate /visibility and the TALPA recommendations for landing.
However, IIRC there have been discussions re rainfall rate and runway drainage involving wet vs contaminated runway braking conditions. Again IIRC a minimum 15 min delay is recommended before landing after heavy rainfall - tower report or red WXR; or a longer time on runways known to have poor drainage, no grooves, or crosswind and other 'damming' effects.
... Then again how does ATC assess rainfall rate?