That said, respectfully, you have much to learn. It's all here, by the way, just dig around for a while.
The Syntec line, with the exception of the German-made 0w-30, is a Group-III based product. It starts as petroleum, and goes through various chemical processes, well beyond simple refining, before the base oil is "complete". Group IV and V oils are synthesized from other materials, such as for example, ethylene gas. Castrol syn blend is a different animal, and not involved in the Castrol-Mobil spat. Syntec has been G-III since the switch, and has not been rebadged a "blend".
The Castrol Mobil thing (an arbitration, by the way, not a lawsuit as its commonly portrayed) arose because until Castrol changed their Syntec line from a PAO (traditional synthetic) to a Group-III base oil, and kept calling it "synthetic", which no one had done before. So while Syntec had started out as a PAO based oil (Group IV) just like Mobil-1, when Castrol had supply problems, and learned that a G-III product could be made cheaper, they made the switch. And, by the way, Castrol won the arbitration, and as I said, they are still sellinig the G-III oil as Syntec.
The problem is that the term "synthetic" has become almost as meaningless as the term "good" or "premium" when applied to motor oil. If you believe that the processing Castrol uses to convert crude oil into Syntec base oil counts as "synthesis", then it's synthetic. If you take a more traditional view of how the word is used with respect to oil, then it most definitely is not. Additionally, the "warring" that has gone on over this has little to do with blends, ironically, though, since many modern SM oils that are thought of as "petroleum" are now actually blends, as they require a synthetic mixed in to meet the demands of the new standards.