The Rif, Rambam and Rosh (and Shulchan Aruch OC 606:1) rule that if one offended his Rebbi he must ask forgiveness until his Rebbi is appeased. The Beis Yosef (DH u'Mah she'Chasav v'Hani) says that the Rif, Rambam and Rosh learn this from Rav, who asked forgiveness from R. Chanina 13 times; R. Chanina was Rosh Yeshiva, therefore he was considered Rav's Rebbi. They explain the Gemara's answer (regarding why Rav requested more than three times) to say that *one's* Rav is different, i.e. one must ask forgiveness from his Rebbi without limit.
My question is that Shabbos 59b and Kesuvos 103b say that Rav was already in Bavel when R. Efes died and R. Chanina became Rosh Yeshiva! My Teretz is that Rav was stringent to consider himself a Talmid because Rebbi had said that R. Chanina would be Rosh Yeshiva (this could also explain why Rav went *only* 13 times), but it is Dachuk in the Lashon of the Beis Yosef. (The Taz explains totally differently, there is no difficulty for the Taz.)
The Lashon of Rashi (87b DH Chelma) "Rosh Yeshiva Hayah" connotes that R. Chanina was already Rosh Yeshiva when he decided to force Rav to go to Bavel. Perhaps Rashi means, he *knew that he would become* Rosh Yeshiva.
Pesach Feldman, Ma'alot Dafna, Yerushalayim, Israel
The Yad Malachi (557) points out that this story took place before Rebbi Chanina became Rosh Yeshiva. See also Neziras Shimshon on Shulchan Aruch (OC 606) who proves likewise.
However, I would suggest that the Beis Yosef does not mean to say that Rebbi Chanina was Rav's Rebbi due to the fact that he was Rosh Yeshiva at the time. Rather, that the fact that Rebbi decided to appoint Rebbi Chanina as his replacement rather than Rav is proof that Rebbi Chanina was greater than Rav. It is reasonable to assume that if Rebbi Chanina was greater than Rav, then Rav probably learned something from him.
As far as Rashi is concerned, I agree with your understanding of Rashi. However, I have a problem with your answer since the Gemara in Sanhedrin 5a relates that Rebbi was still alive when Rav left for Bavel. If so Rebbi Chanina would not yet have been aware of the fact that he was destined to become Rosh Yeshiva.
Perhaps Rav went to Bavel more than once, or Rebbi told Rebbi Chanina that he planned to appoint him Rosh Yeshiva before he told the public at the time of his Petirah. (Obviously this is also Dachuk!)