Birth centres are different from hospitals. I'm Canadian, so I cannot speak to price, but the difference between birth centres and hospitals is that birth centres is that birth centres are equipped with a lot of hospital amenities (no epidurals though) but are less "intervention" prone than hospitals. They're sort of like a middle ground between having home birth and hospital birth. You would have a midwife at a birthing centre. Any birth that's attended by a midwife or obstetrician would be registered as they are required to do a declaration, so the paperwork would be the same. As for the length of stay, it's really individual to a person's needs. Lots of birth centres will discharge within 24 hours, but so will a lot of hospitals now. There's nothing wrong with birth centres or anything shady that would make them more alluring than a hospital, honestly. She may have said birth centre because they're much smaller, so less chance of someone she knows knowing someone in the hospital that she "delivered" at.
I think it's mentioned in the way you've described above because the general population is super uneducated about birth and don't realize that it's a legitimate place to have a baby, just like a hospital or homebirth are.
(Sorry for the novel)