What's is the difference in nuance between visceral (relating to deep inward feelings rather than to the intellect) and emotional?
How do we decide when to use one over the other?
I think the dictionary definitions you are using are a bit misleading in terms of the way we tend to use these words.
Emotional does not necessarily imply any level of intensity, in my opinion. Rather it indicates that you are referring to something that pertains to the emotions, the feelings, the non-intellectual responses to external stimuli. The intensity of the response may be mild, severe, or anything in between.
Visceral, on the other hand, although it also works on a non-intellectual level, is often regarded almost as if it is not connected with the emotions, because it has the connotation of arising from a physical reaction, commonly called a "gut" reaction, a response one feels typically within the abdominal area (commonly: "the stomach"). This is usually felt to be a more powerful reaction, and so we tend to reserve "visceral" for those reactions which are generated by things that hit us so hard that it almost feels as if we have been punched in the stomach.
They are definitely in the same semantic wheelhouse. "Visceral" is related to the word "viscera" meaning the inner guts of your body. So it pertains to something that comes from your gut, your body, your physical inner parts.
Emotional, which physiologically relates to various hormones in your blood, refers to things that provoke a set of specifically emotional responses: anger, sadness, love, passion etc.
Perhaps it is best to think of it in a hierarchy, the degree to which reason plays a role in the action. Consider for example a decision being made by a person, perhaps a woman is trying to decide whether to marry a man.
At the top she might make a rational decision: are we compatible in interests, are our careers compatible, do we have the same goals.
At a second less rational, but still somewhat reasoned level, emotional, she might consider her feelings toward him -- does she have affection for him, does she love him, how does he make her feel?
At a third, even less rational level is her visceral level, is she intensely sexually attracted to him, does she just feel a passion in her gut for him. Not love, but purely from her body. It is her gut, her inner animal, her non rational, non emotional physical reaction.
In my opinion the best way to distinguish between these terms is to recognize that visceral reactions IS a type of deep emotional reaction, the other variant being a deep cerebral emotional reaction. I do not agree with the notion that emotional responses are always non-intellectual. This is a misleading notion.
Thus a visceral emotional reaction is primordial and occurs immediately and spontaneously, whereas
a cerebral emotional reaction is a conscious reaction that involves participation of the intellect (prefrontal cortex) - a learned reaction rather than a gut-reaction.
So visceral is fast, automatic and subconscious, mediated in the subcortical limbic system (so-called reptilian brain) and non-intellectual.
The non-automatic emotional reactions are slow, deliberate and consciously controlled in the prefrontal cortex. From the neurophysiological standpoint, these reactions are intellectual responses.
This is probably extra to requirements, but I feel driven to point it out. In fact, you could more accurately speak of an 'amygdalic' reaction, which covers both 'emotional' and 'visceral'. All information which enters the brain goes first through the amygdala, and reaches the pre frontal cortex about 10 seconds later. The amgydala is what generates the 'fight or flight' response, so you may have an emotional reaction instantly, which can be visceral or otherwise, prior to the information being received in the pre frontal cortex (the thinking part). You may then choose to retain your original emotional response, or reconsider once your thoughts are engaged.
Clearly, then, an emotional response is usually instant, as is a visceral one. But strong emotions do affect body systems, and where we feel it most is the gut (viscera). This is reflected in speech, as in gut wrenching fear, or I feel hollow/hollowed out/empty, which is often said by someone who has their hand on their abdomen.
In other words, it's the strength of the emotion which can make it visceral, so John M Landsberg's response has the right of it, and gets my vote.