Good comments, Abrahm. Here's a compilation of everything we have up to this point:
Amphibian: All living tetrapods (four-legged vertebrate animals) that possess a larval stage, lack protective membranes around the egg, are [ectotherms] and generally spend part of their time on land. The three groups within Amphibia are: [Anura] (frogs), [Caudata] (salamanders), and [Caecilia] (rubber eels).
Urodela/urodele: [caudate].
Caudata/caudate: Caudata or Urodela is the scientific name for the [salamanders]. The word caudate means "tail", as these [amphibians] possess a tail throughout their life.
Salamander: the general term for all caudates, a group of animals that includes the newts, sirens, amphiumas, mudpuppies, waterdogs, olms, hellbenders, etc.
Anura/anuran: The frogs and toads. The word "anuran" means "without tail", as these [amphibians] lack a tail in adulthood.
Caecilia/caecilian: A poorly-known group of amphibians, also known as "rubber eels".
Apoda: [caecilian].
Gymnophiona: [caecilian].
Salienta: another term for [anuran].