No, The first cup plunger was invented as the flushing mechanism in 1777, since the flushing toilet wasn’t invented until almost 1800.
that one looked pretty similar to cup plungers of today.
As far as I can find, accordion and flange plungers were developed later to accommodate the standardized outtake valves of modern toilets.
“Did it apply to older toilets as well?”
Yup, except the first cup plunger was held like a hammer rather than a plunger is held today.
“Were all my parents, relatives, friends parents just cheap and got the wrong one?”
Mine certainly were, and again, this design difference is for some reason not common knowledge, so it’s more likely they just didn’t know that the flanged plunger is specifically made for toilet drains.