i think your getting mixed up with sole custody of the child which concerns Philippine law, while 'sole responsibility' concerns the British embassy - nothing to do with Philippine law, they are 2 separate things.
you could have sole custody under Philippine law but the British embassy might say you don't have sole responsibility. for example say you have sole custody of your child, and you go and live in another part of the phils or in another country but you leave your child with your parents or other family member, years later you decide to apply for a visa for the child, if your parent or who ever looked after the child made the major decisions that effected the child's life and not you, the British embassy might say you don't have sole responsibility, but who ever looked after the child does have. if the father is still in the child life that can be a problem with sole responsibility.