What is the origin of the famous Duke of Wellington portrait? Was it to celebrate him or for a woman he liked?

I know most people like to talk about Napoleon here, but I thought here would be a good place to ask this question.

I am talking about his most iconic painting by Sir Thomas Lawrence, the one everyone sees or thinks about when his name is even mentioned, the one that was once featured on the British £5 note between the 1970s and 1990s.

I thought this piece was from 1815 or 1816 and painted as a portrait to celebrate his victory over the battle of Waterloo. Both Wikipedia and a source from "Age of Revolution" suggest that this painting was commissioned in 1815 and painted and finished in 1816.

But, I have also red from some places that it was originally commissioned by him to Sir Thomas Lawrence in 1817 or 1818 to be painted for Marianne Patterson, an American-origin countess he "greatly admired" and apparently was infatuated with. And he had also commitioned a portrait of hers at the same time. He had given her his portrait while he kept the one of hers in his country home in Hampshire.

I read up on her and Wellington, and what I found so far was that he had met her in 1815 or 1816. And he had potentially had an affair with her, she a married woman herself. Even his neice wrote that he: "fell violently in love with Mrs. Patterson, and it was the amusement of all London." Ironically, Marianne ended up marrying his brother, Richard Wellelsley, later on, which Arthur opposed and was unhappy with. And apparently, she and his brother had a pretty good married life, too. She would keep his portrait until her death in 1853, and it would be given to the 2nd Duke of Wellington.

The differing information is so confusing to me on what exactly is the story behind this portrait and when it was actually painited and where it was before being ending up in his London home Apsley House where it remains to this day. I was hoping someone else had more insight into this.

For reference:

https://en.wikipedia.org/wiki/Portrait\_of\_the\_Duke\_of\_Wellington\_(Lawrence)

https://www.wellingtoncollection.co.uk/collection/arthur-wellesley-1st-duke-of-wellington/

https://ageofrevolution.org/200-object/bregeut-watch-wellington-post-waterloo/

https://www.theguardian.com/culture/2022/apr/03/the-duke-of-wellington-led-men-in-battle-but-preferred-intelligent-women

https://thetravellocker.com/2022/04/13/wellington-women-friendship/

https://ageofrevolution.org/200-object/thomas-phillips-portrait-of-wellington/