Are the qualifications for being called בישול (cooking) the same for Shabbos (e.g., frying is a question in Halacha if it is recognized as Cooking to say אין בישול אחר בישול - there is no such thing as cooking after choking) as they are for being responsible for Cooking meat and milk together, or are they different? Please illustrate with an example.
See the Aruch HaShulchan YD 87:10 where he says that you cannot bring a proof from the laws of Shabbos to say that frying is a problem of cooking meat and milk. Similarly, by smoking (in #25) he writes the same. In #31 he discusses bishul acher bishul, but seems to conclude that again, despite the rules of Shabbos, the rules of basar b'chalav are different.
To some extent it seems to be. See Hullin 108b, Rashi DH "Aherim" and Ritva on that Rashi. He says that he level of Bishul that qualifies as a lav for Basar BeHalav is Ben D'rosai