The severe thunderstorm this morning caused a power outage, so I can’t use the internet right now.
A thunder rain is a rain that's coming with thunder. For all intents and purposes, if you see this, you'll think thunderstorm.
The reading is the on'yomi readings. For 雨, you haven't seen the on'yomi reading, which is う. To remember this, just think about how in a thunderstorm, it rains uni (sea urchin). You go outside, dodging the thunder, to collect these uni.