Generally what appears in caller id is the phone number or something indicating that the caller has blocked caller id - if the recipient's phone system has mapped that phone number to something else, like their name, then their name will display.

At work, I've programmed the phone system directory - so when I call other people in our office, my name shows up on their phones. Since I programmed all the extensions, I could have given everyone any name I wanted and all the phones in the office would use those names.

I've never heard of being able to control what other people see when I call from my call phone though...if it is possible, I would think you'd have to do it through your cell phone provider.