I notice that client names are sometimes shown in other users' project histories. What determines whether or not a clients' name is shown?

The client's name is only shown if both parties agree to display it publicly.

When you enter a project into your project history, you are asked whether or not you prefer that the client's name be shown. If you prefer to display the client's name, an email is automatically sent to the client asking for the client's permission. Once that permission is received, the client's name becomes visible.