Full Disclosure: I read online reviews and went in the movie understanding Arthur would not be a criminal mastermind like Jokers dipected in any other Joker films/comics.
However, being a huge DC fan for over a decade, I must say this sequal is the best dipiction of "The Joker". Allow me to explain. [Spoiler alert below, although I think the movie is still very much worth watching after reading other people's interpretations]
My interpretation of the "Joker", is that it is not a person, but a state of madness. The main character of this sequal is not "the Joker, clown prince of crime", but Arthur Fleck, a kind hearted, weak, mentally ill person. Through a life of hardship and being treated like trash, Arthur was pushed into this state of madness.
Anyone can become the "Joker", and the "Joker" lives in everyone, when being pushed hard enough, anyone can explode and turn mad. [See e.g., Dark Knight, the Joker tries to prove that "Madness is like gravity", when someone is pushed hard enough (have a bad day), they will fall to madness. Joker then drives Harvey Dent, Gotham's White Knight, into the criminal Two-Face].
In this second film, Arthur is not the only person dressed as "Joker": people on the streets don't just put on a clown mask like a typical "Joker Gang" as seen in other movies/comics, they dress the same as "Joker". I understand this to be the director trying to convey exactly this point.
Harley Quinn (Lady Gaga), like many viewers, expect Arthur to become "the Joker" as a criminal mastermind; however, they ignore that "the Joker" is not a person. The "Joker" is a state of being, but Arthur Flek is not constantly in this state of madness. In fact, Arthur wrestles with the madness in him throughout the movie. And just like Harley, many viewers are disappointed when Arthur battled his demons, and chose to not fall into madness, when he had the perfect opportunity (and many would say, well justified) to become an evil criminal.
In my opinion, Arthur is a tragic hero, battling his internal demons. He is kind hearted person, described accurately by his mother Jenny as "he would not hurt a fly". Such a kind hearted person, as seen throughout the sequal, gets treated like trash, and has a seemingly perfect reason to become evil and destroy the city. He was presented that opportunity when the courthouse blew up [probably by Harley]. Yet Arthur, chose to run away from madness [see i.e., running away from random guy dressed as Joker after courthouse explosion]. Harley gave Arthur the opportunity to lead Joker follower in Gotham, but Arthur chose not to go down this path, hence we see Harley's [and some viewers'] disppointment. Harley never loved Arthur; she did not even know Arthur.
The tragegy of Arthur, is that Joker does not end with Arthur winning his battle, because anyone can fall to madness and become "Joker".
This is what I believe the Joker truly is: the opposite of Batman. Batman was once asked in Dark Knight Rises "Why do you wear a mask?", and answers "so anyone can become Batman". Batman stands for order, justice; Joker stands for chaos, madness. They are 2 sides of the same coin. Everyone has a Batman and a Joker inside. It is who we choose to be.