I'm not a historian, but I am an atmospheric scientist and avid mountaineer, so I actually feel like I can answer this one!

First of all, you might be surprised to hear that the invention of the barometer has very little bearing to your question. The barometer was invented in the 1640s and taken up mountains soon after, but long before then there are references to the thinness of air at altitude. In the 5th century writings of Saint Augustine, he refers to the idea that Mount Olympus was so high, that "no wind is felt and no clouds gather, because it exceeds in its height all that vaporous air in which birds fly". Francis Bacon wrote very similar passages in his 1620 work Novum Organum, and furthermore, he attributes this knowledge to "the ancients" who climbed to the summit of Mount Olympus, suggesting that the knowledge goes as far back as ancient Greece (some other texts attribute this to the writings of Aristotle himself).

So did people make the association between thinner air and sickness, or did they attribute it to something else? In the same work as above, Augustine writes that visitors to the summit of Olympus would carry sponges soaked in water, and hold them over their noses for relief, suggesting that they thought the dryness of the air was at fault. However, he also seems to indicate that the dryness of the air was due to its thinness (which in some respects is actually correct!). Additionally, a Spanish Jesuit missionary to Peru named Joseph de Acosta wrote in 1590 about his trip to the high Andes, and describes varying degrees of altitude sickness experienced both by him and others. He attributed the illness to the "subtle and delicate" nature of the air in the high mountains. So it would appear that people did indeed attribute altitude sickness to the fact that the air was thinner at altitude, even without a thorough understanding of air pressure and hydrostatic balance that was discovered much later!

Note: this was by no means universal knowledge; there are many references in Chinese literature as far back as the 1st century B.C. to altitude sickness, even detailed descriptions of deaths by High Altitude Pulmonary Edema (HAPE), but no apparent speculation on the cause. Also, as a scientific note, I'm curious that Augustine mentions a lack of wind and precipitation at altitude; this is counter to what actually occurs, since mountaintop regions are among the windiest and wettest places on Earth (indeed, even Mount Olympus specifically receives 3 to 4 times as much precipitation as the surrounding areas, and often sees high winds). So maybe the observations of Augustine should be taken with a few grains of salt. But since Joseph de Acosta specifically mentions fierce winds, and offers detailed personal accounts to the summits of peaks in the high Andes, I am certainly willing to believe his observations.

tl;dr: Humans have apparently attributed altitude sickness to the "thinness" of the air since antiquity!

Main source: High Life: A History of High-Altitude Physiology and Medicine by John B West; a good number of pages are available on Google Books but subreddit rules apparently discourage linking it and here is the link!