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The expression “free rein” originated with horsemanship, and literally meant that the horse was free from the control of the reins. Use in broader senses followed. “Free reign” became a common substitution, I expect because people weren’t familiar with horsemanship. I’m not sure what “free reign” would really mean. I don’t think royalty pays for the privilege of ruling a kingdom. “Free rain”, well, I think rain is free for everyone.

“To rein in” means to subdue or to control or to limit. “To reign in” would be followed by a place where royals reside and hold power. “To rain in” might be used in a weather report if you are about to say, for example, that precipitation is beginning in a particular place. It is beginning to rain in London.

“To take over the reins” means to take over the leadership of something, but not necessarily the royal throne. “To take over the reigns” would require more than one royal to die or abdicate and leave an empty throne to be filled. “To take over the rains”—I don’t think that’s a thing.