Oh, okay, but what I meant is a "lucky cat" is a lucky cat in this game- if you customize a black one to be white, it's a white lucky cat. If you customize a white one black, it's a black lucky cat. They're the exact same item, but the color is different. And since you can change the color, they're really the same item (I double-checked with the one I keep in my house- you can customize them!). Of course, if you don't have either you'll still need to trade for one, but you won't need to worry about color is my point!
The only one that's different is the GOLD lucky cat, which is a DIY that uses a regular lucky cat as a crafting item- did you maybe mean that one?