In the context of decision making, emotions are those impulses within you that you perceive to "move". Though there are lots of emotions -- anger, hatred, jealousy, joy, lust, etc. -- those emotions come from two basic sources: fear (repulsion) and pleasure (attraction).
By movement, I specifically mean the direction of attention. We tend to avoid paying attention to what we fear, and strive to pay attention to what gives us pleasure. When making a conscious decision, fear (uncertainty, and doubt) is what makes a person hesitate. Pleasure (lust, greed) is what makes a person jump in without regard to risk. You perceive emotions by movement. You have to stand still (mentally speaking), then watch for the movement in the emotions. Because our attention naturally shy away from what we fear, those things tends to get ignored. Because our attention naturally seeks out what we like, we tend to cling to and attach ourselves to those things. When we cling to things we like, we also tend to fear losing them.
Instinct generally refers to survival instincts, and as such are closely tied with emotions.
Intuition, what I think those quotes refer to, don't refer to the absence of emotions so much as being in a state of mind where you are mentally still so you an observe all the movements of emotions. When you observe a particular emotional impulse long enough, it will identify itself to you (that is, you can rationally verbalize that emotional impulse). Once the emotion is identified and heard, they generally pass on and trouble you no more. At some point, you're left with just mental stillness. That's where intuition kicks in.
Just so you know, while some of this sounds like Freudian psychology, where I practice these ideas actually comes from my practice of martial arts. There is not enough time to stand there analyzing the angles and force vectors of an incoming blow, much less formulate a response. That is way too slow. Yet, someone who merely acts just on reflex isn't always able to prevail. There are people with enough trained awareness to manipulate someone's purely reflexive movements. To do well, you have to be able to not only have clear intuition, you have to be able to physically act on it to take advantage of the tiny moments of opportunity. Otherwise, you are unable to seize the initiative and execute on your strategy.
The story passed in martial arts lore is looking into a pool of water: when there are ripples, it is hard to see the reflection. You have to wait for the ripples to smooth over. When the reflection is clear, your image is clear, it looks almost like the real thing. But if it looks too real, and you reach out to touch the image to prove its realness, then ripples form, breaking its clarity. That's the nature of intuition.
I can't say I have successfully applied this to a startup, but maybe you will be able to.