Undefined index: action plugin problem

The logic on line 225 is bad.

if(
  isset($wp_logo_slider_images['update']) == 'Added' 
  || $wp_logo_slider_images['update'] == 'Deleted' 
  || $wp_logo_slider_images['update'] == 'Updated') {

isset($wp_logo_slider_images['update']) is never going to equal ‘Added’ but that isn’t where the error comes in. PHP will try to use $wp_logo_slider_images['update'] twice on that line without checking that the value is set.

What you need is:

if(
  isset($wp_logo_slider_images['update']) 
  && ( 
    $wp_logo_slider_images['update']== 'Added' 
    || $wp_logo_slider_images['update'] == 'Deleted' 
    || $wp_logo_slider_images['update'] == 'Updated'
  )
) {

Notice the extra set of () to group the ||s and make sure that PHP does not try to access $wp_logo_slider_images['update'] if it isn’t set.