The .38 super (and .25 and .32 ACP) are semi-rimmed cartridges. The rim diameter is slightly larger than the case, but they appear rimless to the naked eye. They still headspace off of the rim like a rimmed case. Rimless cartridges headspace off of the mouth of the case.
Similarly, semi-rimed and rimed cartridges head space on the case rim. While a .38 Special can be fired within a .357 Magnum with its bullet farther recessed without problems. The .38 Special pressure curve is much lower than that of the .38 Special.
SAAMI pressure for a .38 Special is 17,000 psi, while ,380 Auto is 21,500 psi and .38 Auto is 26,500 psi. Those other cartridges would be over pressure loads assuming you could get the cartridge to fit in the cylinder.