Purpose of the LN transaction description According to Bolt 11, an invoice may contain a short description. However, what I fail to understand is who is sending this information to whom.

Since the invoice is usually generated by the payee, I'd argue that the short description is some information for the payer, like "Purchase: Product A, Product B". However, I've seen Lapps like tip boxes, where it is the payer who creates the message for the payee.

Or is the idea that the payee could propose a description, but the payer could then change it as he wants?
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.