Tetanus is an anaerobic bacteria which means exposure to air will kill it. If a cut is bleeding that means it's at least partially exposed and may prevent tetanus. The reason people tend to get tetanus infections from puncture wounds is because the wound heals up and seals out the air.
Also, letting a cut bleed for a bit is one way of flushing any foreign material from the wound, which can help. But you still need antibiotics and a bandage to reduce the risk of infection.
B: "Isn't there chest compressions or something involved?"
A: "Nope, chest compressions are dangerous and cause people to break bones. Plus almost 100% of people who were not laying on their backs ended up dying after their heart stops. So, laying someone on their back is all you need to do."
Yeah, it's incredibly frustrating when propaganda uses partially true information to push their agenda. From a first aid perspective, letting the wound bleed a bit and get oxygen exposure can reduce the initial infection. You should absolutely get the vaccine though because why fucking risk it?
Ah right, and that's why people think it's related to rust. Because most people getting a puncture would probably got it by standing on a nail, which was out in the elements and thus rusty…
My doc once explained that nails, or anything that causes a puncture wound, are worse than a cut because they compress tissue within the wound, creating places where there is no exposure to oxygen.