Rides in the rain

What are the criteria for what rides run in some rain and what ones do not.

I am speaking mainly of coasters of course.

Also, why does a water ride shut down at all during rain?

Depends on the material used to construct each ride. Iron Dragon (made of iron) is notoriously fickle in the wet weather, shuts down as soon as a drop appears in the sky. Steel Vengeance (Steel) on the other hand is a bit more resistant. Blue Streak (Wood), gets slick, but the cars stay on the track although it does have a bit more slipping and sliding over the bunny hills. Windseeker does not like or seek wind in any fashion ;)

The initial criteria is set by the manufacturer. Parks can add whatever. For example, Legoland Florida shuts everything down when lightning is unreasonably far away, which means basically every summer day.

Sometimes there are unexpected reasons to close. Wind from a certain direction will close Magnum and Raptor, but they didn't figure that out until a certain gust that stopped the train (for Magnum, in the pretzel I think, for Raptor after the cobra roll).

Cedar Point is notoriously conservative when closing rides for rain. You can travel south to Kings Island and as long as there is no lightning, the coasters will generally continue to operate with all the trains. Meanwhile at CP you can have a literal 30 second mist and they will grind coaster operations to a halt, take trains off, and either have rides closed or running fewer (or single) trains long after the rain has passed.

The lake can create a quick change to the weather so probably more of a caution to shut it down in light rain for what may come.

It all changed after two Magnum trains bumped in the rain in 2007. What many first thought was a temporary overreactive policy has become the norm in the past 18 years.