Could an animal, instead of lungs, have an extra circulatory system that pumps air instead of blood? It seems plausible that it would work with water, as water seems similar enough to blood, but air, as it is a gas, might not work, due to the fact that it is more compressible. Could a single heart pump air throughout an animal, and if not, what must be changed to make it work?
Yes, with a big heart
In terms of scale, your heart pumps around 1,500 gallons of blood every day (source) and your lungs handle around 2,000 gallons of air every day (source). As you point out, air and blood are very different. But if you had a big enough heart, you could pump air through it instead of using a diaphragm with a minimum of hand waving.
To be clear, would your air-heart pump blood into the lungs or replace them completely? If you're trying to replace the lungs, you might want to place restrictions on how physically active your creatures could be. If you look at people pushing their bodies to the limits (like this) you could imagine the pressure closing off some of the air-arteries (airteries?) from all the pressure.
It can be done.
Joseph Pujol, also known as Le Petomane, was able to use his body to basically move air and expel it at wish from one end to the other.
It was a common misconception that Joseph Pujol passed intestinal gas as part of his stage performance. Rather, Pujol was allegedly able to "inhale" or move air into his rectum and then control the release of that air with his anal sphincter muscles. Evidence of his ability to control those muscles was seen in the early accounts of demonstrations of his abilities to fellow soldiers.
Since he was using his muscles to control the air flow and not his lungs, I guess this covers your question.