Asian waving cat Anything and everything

Okay I have been wondering what is the story about Asian waving lucky cat?
Is there many kind of stories about it? Is it coming from China or Japan or both?
I've heard a one story about it.

I never got a clear answer for this so can somebody tell me those stories about that waving cat? =)
Where did it come from?

Thank you!

It's a Japanese lucky charm
That's all I know ^^

Maneki-neko it's a Japanese lucky charm. His hand is beckoning not waving. It was said, that this kind of lucky charm is good for businesses. That's why it was displayed in shops, restaurants and other places like that. Unfortunately I don't know origin of it. Hope it helps :)

I have read that it's also some Chinese thing but mostly people know it from Japan.

I heard story about man who was under the tree and he saw cat like waving to him.
He did go to look that cat and then thunderbolt did hit into that tree where man was sitting.
Cat did save man's life and story did expand all over the country so there did come waving cat =D

But I don't know is this true.