As an appointee you are the representative for your mother,
An AP is an assessment provider HP is the healthcare provider which for DWP is one and the same.
A corporate appointee is usually a private firm or local authority that act on behalf of a claimant in a larger scale and you are an appointee which is an individual appointment for a claimant.
https://www.gov.uk/government/publications/personal-independence-payment-assessment-guide-for-assessment-providers
If that link doesn't work go to the gov.uk website and look for personal independence payment assessment guide for assessment providers. There is pdf download that you can access