
der Bauch
belly
The front part of the human or animal body between the chest and the pelvis, which contains the digestive organs.
Nach dem großen Essen hatte ich Bauchschmerzen.
After the big meal, I had a stomachache.
belly / stomach
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belly
The front part of the human or animal body between the chest and the pelvis, which contains the digestive organs.
Nach dem großen Essen hatte ich Bauchschmerzen.
After the big meal, I had a stomachache.

from the gut
A decision or action that is based on intuition or a feeling, not on rational thinking.
Ich habe keine logische Erklärung, ich habe einfach aus dem Bauch heraus entschieden.
I don't have a logical explanation; I simply decided from the gut.

belly
The rounded, middle or lower part of an object.
Der Bauch der alten Vase war kunstvoll bemalt.
The belly of the old vase was artfully painted.
Pay attention to the case with prepositions. Movement is expressed with the accusative, while a position or state requires the dative.
Der Boxer schlägt seinem Gegner in den Bauch.
The boxer hits his opponent in the belly.
The idiom 'aus dem Bauch heraus' (from the gut) is very common and means that you make a decision intuitively. The expression functions in the sentence like an adverbial phrase.
Ich kann es nicht erklären, ich habe einfach aus dem Bauch heraus gehandelt.
I can't explain it; I simply acted from the gut.
In connection with the body part, 'Bauch' (belly/stomach) is often associated with pain or feelings. Typical phrases are 'Bauchschmerzen haben' (to have a stomachache), 'Schmetterlinge im Bauch haben' (to have butterflies in one's stomach), or 'sich den Bauch vollschlagen' (to stuff oneself).
Vor der Prüfung hatte ich immer ein flaues Gefühl im Bauch.
Before the exam, I always had a queasy feeling in my stomach.
When 'Bauch' (belly) is used in connection with objects such as bottles, vases, or ships, it refers to the rounded, bulging shape of the object. This is a figurative transfer.
Der Wein lagerte im Bauch des alten Holzfasses.
The wine was stored in the belly of the old wooden barrel.
stomach
Bauch refers to the entire front part of the body between the chest and the pelvis. Magen is a specific digestive organ in the upper abdomen. Bauch fits general or imprecisely located discomfort; when the organ or its digestive function is meant, Magen is the right word.

Nach dem Sturz tut mir der ganze Bauch weh.
After the fall, my entire abdomen hurts.
Nach dem schweren Essen tut mir der Magen weh.
My stomach hurts after the heavy meal.