Wordpress - Creating a wp_editor instance with custom tinyMCE buttons

You pretty much had it, according to the description.

Here's what you might be looking for for instances 2 and 3 (for instance 1 you can leave the settings empty to get the default set of buttons):

Instance 2:

wp_editor(
    $distribution,
    'distribution',
    array(
      'media_buttons' => false,
      'textarea_rows' => 8,
      'tabindex' => 4,
      'tinymce' => array(
        'theme_advanced_buttons1' => 'bold, italic, ul, pH, temp',
      ),
    )
);

Instance 3 (showing each of the 4 rows you can set for TinyMCE):

wp_editor(
    $distribution,
    'distribution',
    array(
      'media_buttons' => false,
      'textarea_rows' => 8,
      'tabindex' => 4,
      'tinymce' => array(
        'theme_advanced_buttons1' => 'bold, italic, ul, min_size, max_size',
        'theme_advanced_buttons2' => '',
        'theme_advanced_buttons3' => '',
        'theme_advanced_buttons4' => '',
      ),
    )
);

I recommend that you check out the wp-includes/class-wp-editor.php file (specifically the editor_settings function on line 126) in order to understand how WP parses the settings you use inside the wp_editor() function. Also, check this page to understand more about the functionality of TinyMCE and its init options (which I don't believe WP support fully).

you can set the params via array on the wp_editor() function; an exmaple

$settings = array(
    'tinymce'       => array(
        'setup' => 'function (ed) {
            tinymce.documentBaseURL = "' . get_admin_url() . '";
        }',
    ),
    'quicktags'     => TRUE,
    'editor_class'  => 'frontend-article-editor',
    'textarea_rows' => 25,
    'media_buttons' => TRUE,
);
wp_editor( $content, 'article_content', $settings ); 

You can set the values via array in the param 'tinymce', 'tinymce' => true, // load TinyMCE, can be used to pass settings directly to TinyMCE using an array() Also it is possible to get about the params of buttons: theme_advanced_buttons1, theme_advanced_buttons2, theme_advanced_buttons3, theme_advanced_buttons4

array( 'theme_advanced_buttons1' => 'bold, italic, ul, pH, temp' )

also you can get via filter hook to create custom buttons, also an example

function fb_change_mce_options($initArray) {
    // Comma separated string od extendes tags
    // Command separated string of extended elements
    $ext = 'pre[id|name|class|style],iframe[align|longdesc|name|width|height|frameborder|scrolling|marginheight|marginwidth|src]';
    if ( isset( $initArray['extended_valid_elements'] ) ) {
        $initArray['extended_valid_elements'] .= ',' . $ext;
    } else {
        $initArray['extended_valid_elements'] = $ext;
    }
    // maybe; set tiny paramter verify_html
    //$initArray['verify_html'] = false;
    return $initArray;
}
add_filter( 'tiny_mce_before_init', 'fb_change_mce_options' );

also you can filter the buttons directly; each line have an each filter: mce_buttons, mce_buttons_2, mce_buttons_3, mce_buttons_4

the follow params are the default for the example on hook: tiny_mce_before_init

'mode' => 'specific_textareas'
'editor_selector' => 'theEditor'
'width' => '100%'
'theme' => 'advanced'
'skin' => 'wp_theme'
'theme_advanced_buttons1' => 'bold,italic,strikethrough,|,bullist,numlist,blockquote,|,justifyleft,justifycenter,justifyright,|,link,unlink,wp_more,|,spellchecker,fullscreen,wp_adv'
'theme_advanced_buttons2' => 'formatselect,underline,justifyfull,forecolor,|,pastetext,pasteword,removeformat,|,media,charmap,|,outdent,indent,|,undo,redo,wp_help'
'theme_advanced_buttons3' => ''
'theme_advanced_buttons4' => ''
'language' => 'de'
'spellchecker_languages' => 'English=en,Danish=da,Dutch=nl,Finnish=fi,French=fr,+German=de,Italian=it,Polish=pl,Portuguese=pt,Spanish=es,Swedish=sv'
'theme_advanced_toolbar_location' => 'top'
'theme_advanced_toolbar_align' => 'left'
'theme_advanced_statusbar_location' => 'bottom'
'theme_advanced_resizing' => true
'theme_advanced_resize_horizontal' => false
'dialog_type' => 'modal'
'relative_urls' => false
'remove_script_host' => false
'convert_urls' => false
'apply_source_formatting' => false
'remove_linebreaks' => true
'gecko_spellcheck' => true
'entities' => '38,amp,60,lt,62,gt'
'accessibility_focus' => true
'tabfocus_elements' => 'major-publishing-actions'
'media_strict' => false
'paste_remove_styles' => true
'paste_remove_spans' => true
'paste_strip_class_attributes' => 'all'
'wpeditimage_disable_captions' => false
'plugins' => 'safari,inlinepopups,spellchecker,paste,wordpress,media,fullscreen,wpeditimage,wpgallery,tabfocus'

see on this link for more infos to this filter.

Just to update this as I had to dig in the wp source files

$settings = array(
    'tinymce' => array(
        'toolbar1' => 'bold, italic',
        'toolbar2' => '',
    ),
    'wpautop' => false,
    'media_buttons' => false,
);

I think this had changed with Tinymce 4.