Most farts are the result of bacteria in your large intestine feasting on undigested food that your body was not able to break down and absorb.
As an example, beans contain 2 types of sugars which you people do not have the enzymes to digest/break down/absorb. Bacteria do have these enzymes, so they can break down these sugars. Bacteria can also digest plant material which you can not. A by product of the bacterial action is gas.
The vast majority of farts are NOT the result of swallowing air.
I (and many others) have tested farts. Air is primarily nitrogen gas, but most farts contain very little nitrogen gas so they are not due to swallowed air. Most farts are made of carbon dioxide. CO2 is a byproduct of bacterial action.
Hydrogen sulfide is present in farts in smaller amounts, and this gas gives farts their bad smell. Unabsorbed protein that is digested by bacteria results in smelly farts.
Theories on why silent farts can be smellier.
A high protein meal may not have roughage and fiber. Fiber increases intestinal motility and transit time. Without much fiber, protein meals move through the intestine more slowly and bacteria have a chance to break them down more completely
The result is relatively more smelly gas.
Higher fiber meals with more plant roughage move through the intestine more quickly. Less chance for any protein to digest so relatively less smelly gas.
High fiber plant diets result in higher volumes and percentages of carbon dioxide being produced. This results in higher volumes of gas and thus higher volumes of farts. With lots of CO2 in a fart, the relative volume of smelly gases is diluted and less noticeable.
It is likely you produce the same amount of stinky gas during one or two lower volume 'protein farts' than you do in 5-10 regular farts. The gas and smell is diluted and spread out in numerous regular farts, and thus not as noticeable.
Also noisy farts are the result of your anus effectively flapping in the breeze.
Higher volumes of gas in regular farts results in a greater likelihood of increased noise. Lower volumes of gas in protein farts more likely to have lower gas volumes and less noise. Thus silent farts more likely to be stinkier
(silent but deadly).