MDMA is the chemical name, like ethanol is the chemical name for alcoholic beverages.
My understanding had always been the two were synonymous.
This is a common misconception. MDMA is often presented as "pure" (often by the people selling it), but in reality, almost all MDMA outside of a pharmaceutical setting is impure. Most MDMA contains some amount of methamphetamine[0], but not necessarily enough to be subjectively identifiable by a user as "meth".
Also, the half-life of MDMA is longer than cocaine, but shorter than methamphetamine.
[0] MDMA is actually a methamphetamine, but it's not the one that people are referring to when they say "meth" colloquially.