#2 makes sense. "But" here functions as a conjunction joining two complete phrases. It really splices two sentences together. #1 does not make sense because "but" has no plausible role. It is not a conjunction because the second phrase is not a complete sentence. It doesn't make sense as an adverb because what would it modify? Nor does it make any sense as a preposition. A good strategy would be to express your thought using two sentences.

I'm not sure if #1 is technically incorrect or not, but it sounds strange to me. It's the "but thinking" that sounds unnatural. May I suggest, "You don't want to be too fixated on winning but (instead) think about learning from losing/your losses." If you want to maintain the structure of not-fixated-winning/but-fixated-learning, then I suggest adding the preposition "on" like this, "You don't want to be too fixated on winning but on thinking about learning from losing." #2 sounds about as natural as it can aside from my next observation that covers both. For both of them, using "but" to connect these thoughts doesn't sound quite natural to me. Instead, I would break each one up into two sentences and I'd probably use "rather", like this: "You don't want to be too fixated on winning. Rather, think about learning from losing."

Yes, both of these sentences are grammatically correct and sound natural. They convey a similar meaning, which is that winning is not the only important thing, and that learning from losing can be valuable. However, you may want to consider changing "thinking about" in the first sentence to "learning from" to match the second sentence. This would make the sentences more parallel in structure and reinforce the idea that the focus should be on learning from losing rather than just thinking about it. Here's an example of the revised first sentence: "You don't want to be too fixated on winning, but rather focus on learning from losing."