In my experience, when I send money from my current account into my daughter's using telephone banking, I am asked to give the sum a name. Generally I get the assistant asking, "Will she know who it's from or do you want to give it a title?", resulting in me saying something like "Oh, can you put 'flowers' please", so she'll know it's for the flowers she ordered on my behalf".
Sure enough her statement shows TFR Flowers £50 (TFR = transfer). If I didn't give the transfer a name it would simply show her account number plus the sum.
This suggests to me that your benefactor could either just leave the details showing nothing but his/her account number or, if they feared being recognised by that, could opt for a description, e.g. "Church roof", or "donation".
If I make a transferal into DD's account online it will show up as the sum plus my account number so if your benefactor wants to avoid that, a call to his/her bank is the way to go.