I simply can't suspend disbelief enough to make the letters from Delphine. No matter what the wiki says (which, after all, is NOT canon and truth - it's only what players state/edit....)
Good cause you don't have to suspend disbelief cause she's the one who sends you the letter if you don't see her talking to the mage in whiterun, go back and read the letter that she put in the tomb when looking for the horn, its signed by a Friend as well. You will only get the letter (the first one after killing the dragon) if you have the Dragon Stone from Bleak Falls Barrow before talking to the Jarl's mage in whiterun (who asks you to go to Bleak Falls to get it). Now if you go to Bleak Falls Barrow After talking to the mage and return with the dragon stone you will see her there talking to the mage, and after dropping the dragon at the watch tower you will not recieve the letter.
I've tested it out on 5 characters, and the only ones to recieve the letter from "A Friend" after killing the dragon at the Watch Tower, were the ones who cleared out Bleak Falls Barrow before talking to the Jarl of Whiterun's mage and turned it in to him when you first meet him. The ones who did not clear out Bleak Falls Barrow before the mage asks you too , never recieved the letter "From a Friend" and instead meet Dalphine when see is talking to the Mage about his dragon research upon returning from BLeak Fall Barrow.
If you disagree with me please test it out yourself or if you need to see how to activate the scenes please message me and I will send you a more detailed walkthrough