Some personal observations:

Reacting and Thinking are definitely two separate mental modes for me. I can do both fine, but I prefer not to mix the two. When I'm in the thinking mode, I cringe when I have to react on stuff like email or errands or coworkers' requests. When I'm in the reacting mode, I tend to avoid thinking-type problems that require uninterrupted time.

I often face this distinction at work. I'm a lead developer (not a coder) and, simultaneously, a project manager of a relatively large software project, so I often have thinking-type stretches of work (say, designing a major feature or subsystem for the next version of the software) followed by reacting-type periods when I oversee the implementation. Mixing the two kinds of work is painful for me, so I always try to separate them.

For thinking-mode periods, I prefer to shield myself from interruptions. For reacting-mode periods, I make myself open for interruptions, and even actively seek them.

Switching between the modes takes time, usually several days.

As for safely training oneself to react to external circumstances in real time: I definitely think that competitive / team-based first person shooters might be beneficial here, based on my experience with Team Fortress 2.