Came up with this gem last night

Not really. Sort of? Both are expelled had that cannot be otherwise absorbed by the body, but Farts come from the intestines while a “regular” burp comes from the stomach. It is possible for a fart held in or unable to escape through the anus, to either enter the blood and be exhaled through the lungs, or to literally bubble up and come from your mouth. This second type would be technically a “fart from your mouth” I suppose. But most of the air in a burp is swallowed air trapped above the plyoric sphincter, where as most fart gas is the byproduct of decay and other chemical reactions with food and waste.

Farts are just butt burps.