Messages - setting my name

If you're using an iMessage account, the sender doesn't have much control (you can't create a nickname/alias, in other words). If you and the recipient are using another account type that allows for a nickname (jabber, AIM, Google, or Yahoo) , then your nickname may be displayed.

On the recipient's end, the Messages app searches the Contacts app for a contact card that matches the incoming sender's account name/alias. If there is no match, the recipient sees the account name/alternate account name of the sender. If there is a match in Contacts (a contact card with an email address, AIM name, phone #, etc. that matches the incoming message), then the first and last name on that contact will be displayed.

If a recipient sees an account name/alternate account name as the sender, it's a hint to add that account name to their Contacts, either as a new contact card, or added to an existing contact card. They can do that directly from the Messages app.

In iMessage, the sender does get to choose what account name/alternate account name they use when starting new conversations. You'll find that in OS X Messages > Preferences > Accounts > iMessage account > Settings: Start new conversations from. In iOS Messages go to Settings > Messages > Send & Receive: Start new conversations from. The choices are limited to iPhone phone #, Apple ID, and Apple ID alternate email addresses.