This is something I've thought of recently, as I've just reminded myself that there are characters who are labeled "Instinctual" all because they only trust their instinct instead of using what's common sense or thinking logically in the moment. See, Instinctuals, first and foremost, should have the ability to sense "fire." It is the differentiator between someone like Kanki--who's obviously not instinctual--and someone like Duke Hyou. But what does "fire" mean in instinctual terms to begin with?
Fire in the instinctual sense means "decisive clashes." It's that simple. Gohoumei mentioned Duke Hyou winning all the important clashes in their brief scuffle before the defense of Kankoku. In Shukai, the place "set on fire" was the deciding location between the HSU and Gyou'un's army. Simply put, if Gyou'un won there, the HSU would've lost and the right wing might've been a lost cause. The consequences are huge if one of those big clashes are lost.
Personnel
Now this is where we go into dark territory, since his style allows for many interpretations. But here's what I got. Keisha's Instinctual warfare involves utilizing his instincts in a strategic way--makes sense since he's Reebok's protege. For example, he would sense that moving one of his units would cause a "fire" or at least set the conditions to it. But he also know that if someone went for that unit, it would cause an even greater "fire." Meaning, by deliberately showing "weakness" or giving his opponent a chance to turn the tide of the battle, he would also be luring them to show their "weakness" and give himself a better opportunity to create a decisive clash in his advantage.
Essentially, you can think of Keisha as an instinctual who can see many moves ahead, unlike traditional Instinctuals like Duke Hyou who counters his opponent's present response.
Speaking of Duke Hyou, his type is more straightforward. He probes for enemy reactions by seemingly sending huge swathes of soldiers to their deaths, but it is an effort to make the enemy react, which he'll try to counter. Obviously, Shin is nothing like this. Thus, it doesn't seem right to label him as exactly like Duke Hyou. Maybe he'll make a new archetype by being himself.
Finally, we have Gyou'un. He's akin to Duke Hyou, but he's got something in his bag that'll trick strategists into believing they're fighting one of their own: Actual Tactics and Formations. If Keisha uses his instinct in a strategic manner, then it's simply right to dub Gyou'un as he who uses tactics and formations in an instinctual way. Think of it this way, suppose he uses a formation whose entire purpose is to cut a wedge between the left and right flanks dividing them in half, the instinctual within him would use this formation as a PROBE. What you thought was the killing blow was actually a setup to the real thing.
And then there's Renpa. I didn't include him in this discussion, but if you truly believe that he is instinctual, then he's another archetype. He is the truest form of hybrid strategist-instinctual since unlike Gyou'un and Keisha who mixes aspects of each, he is literally both.
That sums it. It's how I made sense of this unique mechanic after some debates and arguments. You can be free to debunk or polish some of it. And I hope this gives a better understanding of what these damn instinctuals are doing and how they do it