i386 or amd64 debian for new intel 64 bit cpu?

I've got a new intel 64 bit PC. I installed i386 latest debian on it.

It works fine. Then I noticed there is amd64 debian also.


Would it be better to install amd64 instead? I am assume amd64 will work on this pc. How is debian 64 bit support for 32 bit applications? Why is it called amd64 btw? why not intel64? may be becuase amd had 64 bit before intel?

I have to format the disk and start all over, I think, but that would be ok.