I hope others clarified it for you, but in case your question to me is still hanging, this is a longer version of my comparison of the two men's claims:

Wickham, Ch. 16:

"...My father began life in the profession which your uncle, Mr. Philips, appears to do so much credit to -- but he gave up everything to be of use to the late Mr. Darcy, and devoted all his time to the care of the Pemberley property. He was most highly esteemed by Mr. Darcy, a most intimate, confidential friend. Mr. Darcy often acknowledged himself to be under the greatest obligations to my father's active superintendance, and when, immediately before my father's death, Mr. Darcy gave him a voluntary promise of providing for me, I am convinced that he felt it to be as much a debt of gratitude to him as of affection to myself."

Here Wickham says that the elder Mr. Darcy gave the elder Mr. Wickham a voluntary promise immediately before the elder Mr. Wickham's death. This may not have been a "deathbed promise," but that is the simplest idea that fits the facts.

Then Darcy writes, less melodramatically (Ch. 35, of course):

"My excellent father died about five years ago; and his attachment to Mr. Wickham was to the last so steady, that in his will he particularly recommended it to me, to promote his advancement in the best manner that his profession might allow -- and if he took orders, desired that a valuable family living might be his as soon as it became vacant. There was also a legacy of one thousand pounds. His own father did not long survive mine..."