Thinking  is very poorly understood, but broadly speaking in seems to involve the prefrontal cortex (PFC), with a special role for the dorsolateral PFC. Other important areas include the hippocampus and parietal cortex. Ultimately, thinking involves many brain regions, and cannot be localized to one place. Thinking is a distributed process that can incorporate many different parts of the brain. And the specific content of the thoughts will influence which brain areas are involved. If you are thinking about images, visual areas will be involved. If you are thinking about movement, motor areas will be involved.
In my opinion, the distinction between “linear” and “creative” thinking is somewhat vague. At this point, the most important thing to note is that the idea that the “left brain is rational/logical and the right brain is creative/artistic/emotional” is totally wrong  . Both hemispheres contribute to logic as well as creativity. Moreover, the use of logic can itself be a creative activity.
There are two ways to define thinking: each leads to a different answer to the question of whether thinking is conscious or not.
Thinking as a subjective experience. If someone asks you what you are thinking about, you can introspect, and describe your thought process. You can also say that you weren’t really thinking at all.
Thinking as the cause of ideas and thoughts. If you discover a thought, then you can infer that the process that led up to the thought was a form of thinking, even if there was no subjective experience associated with the process.
We can test our preference for definition 1 or definition 2 by considering an example.
Sherlock Holmes was a good chemist. When he found himself stuck while attempting to solve a mystery, he would sometimes distract himself by doing a chemistry experiment. At the end of such an experiment, he often found that a solution simply popped into his head.