Luke: So I am a Jedi
Yoda: (laughing) No. One thing remains. Vader. You must confront Vader. Then, only then, a Jedi will you be.
Ya...but as far as his training goes, he was a jedi knight and he did confront Vader so yes he was a jedi knight.
Luke was trained, yes. Fully trained is a matter of debate. Yoda and Obi-Wan didn't have near the time it would take to fully train him. Self taught? He would have to do some major communing with the Force and then get Lightsaber techniques downloaded into his brain, imo. At least it wasn't Keanu Reeves saying "Dude, I know Shii-Cho! Most Excellent!" Although, tbh the originals COULD have had better swordfights. That was the one thing the prequels pwned the OT on.
No he was fully trained, the novelizations of the movies and between the movies go into far more detail of Luke's training. You can only shove so much into a movie, books you can do whatever you want without any limits.