The second and third movies of Twilight were still about Iza and Edward. The main plot and ending of the two movies were very similar, especially the third movie, which almost completely repeated the plot of the second movie.
Edward in Twilight fanfiction is typically depicted as having an intense internal struggle. His love for the human girl is always at odds with his vampire instincts. He may be more conflicted in fanfiction than in the original books, as authors explore different scenarios. For example, some fanfics might show him having to choose between his family and his love in a more extreme way. Also, his relationship with other vampires can be further developed in fanfiction, showing his loyalty or lack thereof in different situations.
One great fanfiction could be where Edward and Bella's relationship is explored more deeply in a different vampire society setting. For example, they might be in a vampire clan that has unique rules about human - vampire relationships.
In many Twilight fanfictions, Edward and Bella's kiss is often depicted as an extremely passionate and intense moment. Some fanfictions might explore different scenarios around this kiss. For example, it could be set in an alternate universe where they meet under different circumstances before the kiss. The kiss in fanfiction can also be used to show a deeper connection between them than what was shown in the original books. It allows the writers to play with the emotions and the relationship dynamics in a more creative way.
There's also 'Twilight Reimagined: Edward and Jacob'. In this one, the author completely reimagines the story. Edward and Jacob are forced to work together in a world where the Volturi are more powerful than ever. It has a lot of great character development and the plot twists keep you on your toes.
One great 'Twilight Edward Story Fanfiction' is 'A Different Shade of Twilight'. It offers a fresh take on Edward's character, exploring his internal struggles in a new and interesting way.
Another common theme is the exploration of Edward's vampiric nature. Fanfic writers might focus on how he deals with his thirst for blood while being in love with a human, or how his vampiric abilities influence his relationships.
Maybe it's because of a past traumatic event related to water that happened to him. Or it could be a supernatural reason specific to the fanfic's alternate universe.
A common theme is the 'what - ifs'. What if Bella had chosen Jacob instead? Fanfictions often explore this alternative storyline and how it would affect the vampire - human relations in Forks. It can be really interesting to see how the characters develop in this different scenario.
On Fanfiction.net, you can use the advanced search feature. Look for stories tagged with 'Edward's perspective' and 'Twilight'. There are often some really good ones that delve deep into his thoughts and feelings, which can add a whole new dimension to the Twilight story.