NeverarGreat said:
by the end of the OT, Anakin is definitely Obi-wan’s apprentice:
“When I first knew him, your father was already a great pilot, but I was amazed at how strongly the Force was with him. I took it upon myself to train him as a Jedi. I thought that I could instruct him just as well as Yoda. I was wrong.”

‘Train’ and ‘Instruct’ are rather strong words to use about someone you just hang around with.

Perhaps Obi-Wan wanted to train Anakin properly, but Anakin got distracted by other priorities (mainly his love for the future mother of his children) and it didn’t work out. So they part ways amicably and Obi-Wan departs to search for other talented novices, as the final credits roll.

Another issue that needs to be settled: if we dismiss the idea of a centralized Jedi order, what really makes a Force user into a Jedi knight? Perhaps Anakin could be knighted for his services to Alderaan by king Organa shortly after he began learning about the Force. Obi-Wan himself received the very same honour at some earlier point in time. Shortly after that, a conversation such as this one could take place:

Very well, Anakin, you are now a Jedi Knight.
A Jedi? Hardly. I still have too much to learn.
Don’t we all?

This ties in nicely to the OT:
“I was once a Jedi Knight, the same as your father.” Obi-Wan considered Anakin almost a full-fledged Jedi Knight, while Anakin felt he needed more training to earn the title. Vader’s dialogue further supports this notion: “When I left you, I was but a learner. Now I am the master.” and “The Force is with you young Skywalker. But you are not a Jedi yet.”

On the other hand, Luke seems to be a lot less diligent than his father: “I can be a Jedi. Ben, tell him I’m ready. I’ve learned so much.”