I would use $backups_to_delete for the loop to avoid creating another variable ($count) for readability reasons. Also, let's break early:...

I would use == 0 because is clearer in the sense of 0 means no limit, and I guess negative numbers are not allowed, so only 0 would match < 1.

I would recover the variable $number_of_backups so you don't run count($back_list) twice. Not of performance reasons but for readability. Also, I w...

minus -> less