To display the button on the rows, you need to put the attributes in the typeAttributes object just like with actions in the row above in your example.

{
  type:  'button',
  typeAttributes: 
  {
    iconName: 'utility:edit',
    label: 'Edit', 
    name: 'editRecord', 
    title: 'editTitle', 
    disabled: false, 
    value: 'test'
  }
}

Update

The above example shows just how to add the button to the column to display it. Manza rightly asks how to use the button to call a method.

First, you add a method to your controller to handle the button click. Something like this:

handleRowAction: function (cmp, event, helper) {
    var action = event.getParam('action');
    var row = event.getParam('row');
    switch (action.name) {
        case 'editRecord':
            helper.editRecord(row);
            break;
        // You might have other buttons as well, handle them in the same way
        case 'action2':
            helper.handleAction2(cmp, row, action);
            break;
        // etc.
    }
}

You retrieve the value of the name attribute you defined in your column using the event parameter. Based on this you can execute different logic as shown in the example above.

Then you add a onrowaction attribute to the lighting:datatable component:

<lightning:datatable
                columns="{! v.columns }"
                data="{! v.data }"
                onrowaction="{! c.handleRowAction }"
                />

This will trigger the handleRowAction method you defined in your controller when a button on a row is clicked.

Also note that if you have buttons in columns as well as row actions in the action column (the little drop down menu), these actions when clicked will also fire the same onrowaction method, so be sure to handle those in that method as well.

Please see the Data Table in Action example in the component documentation for further example code and documentation.

Yes, it only works on the checkboxes. Unfortunately, there is no onrowclick type handler as there was in Visualforce right now. If you want to make a Lightning Data Table that has this feature, you're going to have to write your own component(s) to do this.