To properly debug this you might use your Browsers Console! Especially the "network" tab is a great help in debugging this.
You included the scripts successfully, used the correct markup, so that the DOM gets changed. Now the next step in debugging is, to ensure that the ajax request (pushing the data) to your post.php is made.
To send something you need to set ajaxOptions put, like this:
$('#name').editable({
type: 'text',
url: 'post.php',
ajaxOptions: {
type: 'put'
}
});
You debug the Ajax request by checking the "network" tab of your console. Open the network console before you edit the field. Then edit it and watch the new request appearing in the console. By clickling the log entry in the console, you see the data send from the browser to your script.
In your post.php you might add a var_dump($_POST); to see all the parameters incoming.