I was checking my version before reporting a different bug, and cleared the 'rain' weather quickly before exiting the world, then came back on after about a minute and it was odd light rain, but without they greyed skies. After leaving Minecraft on for a bit while creating this report, it had stopped.

Photos: last one: the original glitch; second to last second one: a reproduction of the bug, but with a thunderstorm (the mobs don't die, which would be bad for survival, unless if you wanted a thunderstorm while you could still see well); third to last one: normal rain; fourth to last one: a normal thunderstorm.

Update: making things clearer.

To reproduce: 1. do make it rain or thunder, either naturally or with commands.

2. Do /weather clear or wait for it to start clearing naturally.

3. Exit the world after a few seconds of putting the second command in, then come back on.

You will see some weird things, eg the skies being blue and only having very light rain, or having rain but mobs that wouldn't normally burn in rain being damaged, but without the on fire effect.

1.18.2 update: I can't get it to do the same things, but I can get it to do some similar things. Eg, if I set it to lightning then quickly leave the world, channeling, hostile mobs spawning, etc won't work, but it will be raining. Or, if I do the opposite (have it on thunder then clear and leave), then the same thing will happen.
However, the sky isn't as weird as it was pre-update.