First, "an historian" is a common usage; no need to get mad about it. Get angry about "based off" instead, because it makes no sense.

In answer to your main question, I cannot speak to China or Mesoamerica, but yes, until at least the late 18th century, the dominant way to understand digestion was that food provided fuel for the "vital flame" inside us. They did not--and neither does Laudan, in the article--claim that these were literal flames, but physicians did certainly recognize the importance of food and bodily heat. The idea of vital flame that maintained bodily heat and life is a very old one, going back to Hippocrates, if memory serves. They not illogically connected the heat of the cooking fire to the heat of the body.

The change begins with the shift toward biochemical understandings of the body, beginning in the late 18th century. Antoine Lavoisier's work was crucial in this, though he was killed in the French Revolution and his career was cut short. Still, his work in identifying the importance of oxygen in combustion--and the heat that was generated thereby--was an important moment in the history of nutrition. In the 19th century, the focus of doctors and physiologists studying digestion followed Lavoisier's lead and sought to better understand how combustion and digestion were connected. There was a broad consensus that digestion was a form of bodily combustion. So, yeah, I agree generally with Laudan that most people pretty much thought of cooking fires and internal digestion as quite similar.

I should also note that Laudan is fairly well thought of. I own at least one of her books, and Gastronomica, while not the most rigorous publication, is still a University of California peer-reviewed journal.

From a biochemical standpoint, that's not far off. Bioavailable energy is obtained through oxidation of high energy macromolecules (sugars and fats, primarily, although proteins will do in a pinch), which is the same basic chemical process as setting a block of wood on fire.