Purpose of the LN transaction description
I believe it is to explain the "purpose of the payment" as described in BOLT11:


  Short description of purpose of payment (UTF-8), e.g. '1 cup of
  coffee' or 'ナンセンス 1杯'


This would be the payee letting the payer know what they are paying for, e.g. in case there are multiple outstanding invoices, just like any normal invoice outside of LN. Like you said, it's perfectly reasonable for the payer to suggest a description, and would probably be preferred in some cases, so the payer knows the payee has agreed to the purpose of the payment and is expecting it.