Is it still called "The Bean"? Or is it now called "The Cloud Gate"? Or both?

It's both. Cloud gate is the official name. The bean is the nickname

Da Bean

Anish Kapoor designed this project and called “The Cloud Gate”... people have just given it the name “The Bean” and the funniest thing is, Kapoor hates it being called a bean.

which do you think is going to get claimed first?

Both

One is official, one is not official.

Warren is correct. It has always been that way.