When dealing with themes, it's general practice to show the user's name as the value clearly labeled as "Display name publicly as":

The default JSON response for embedded users (e.g. authors in posts) includes all of these values:

{"author": {
  "ID": 4,
  "username": "kadamnation",
  "name": "K. Adam White",
  "first_name": "K. Adam",
  "last_name": "White",
  "nickname": "KAdam",
  "slug": "kadamnation",
  "URL": "http://www.mydomain.com",
  "avatar": "http://0.gravatar.com/avatar/6cd1cf6de05952505a165425efaf06b4?s=96",
  "description": "Some short biographical description about the user",
  "registered": "2012-06-17T15:41:32+00:00",
  "meta": {
    "links": {
      "self": "http://www.mydomain.com/wp-json/users/4",
      "archives": "http://www.mydomain.com/wp-json/users/4/posts"
    }
  }
}}

Of these, we discussed at WCNYC that we may want to limit the information about a user that we expose. I propose limiting embedded response data (and responses for non-authenticated requests, see #297) to these values:

• name: (the name explicitly selected for display on the site)
• slug: The public slug representation of the user, for use with routes etc
• nickname: If this gets populated, I think it'd make sense to expose it since I've seen it commonly used on things like author archive pages

This would mean we would omit the following:

• username: Even though this is probably the same as slug (need to confirm this), we probably don't want to give away internal account names where possible
• first_name: Superseded by name
• last_name: Superseded by name
• We also discussed removing registered, unless anyone can think of a compelling reason why it should be in the public responses