To answer your question of the headline first: You are searching for the German word Doppelpartikelverben (in English, you could use double phrasal verbs or double particle verbs, but as they don’t exist in the English language, there is no official term).
Some precision to your question (and this answer)
Your explanation and comparison of prefixes is only half correct as you assume that all syllables before the verb are prefixes, but some of them are called particles (in German Partikel).
Prefixes are non-dividable and aren’t words for themselves (as a big difference to particles; e.g. be-, ver-). Linguists assume that they originate from prepositions.
Particles are words, which are also real German words for themselves (e.g. über, durch, which can also be prepositions and adverbs; über is is in addition to this adjective).
What you called inseparable prefixes can be prefixes in the narrower sense (Präfixverben or prefix verbs) as a verb with a prefix. But they can also be verbs with a particle in front of the verb (Partikelpräfixverben; in English, I would suggest particle-prefix verbs). The particle of the latter ones are treated as a prefixes (complicated, huh?).
Both compound verbs have their phonetical accent on the verb itself (and not the particle/prefix).
They are called phrasal or particle verbs (Partikelverben in German) and have a particle in front of the verb.
If the word has two particles, then it's called Doppelpartikelverb. You can’t find words with more than two particles in the German language. But not all these verbs are separated into three single words in the verb second word order. (Maybe, this is historical, because people are lazy and forgot the space between the two particles over the decades, but this is a blind guess.)
The phonetical accent is on the particle (the second, if you have two).
Prefixes that can be either separable or inseparable
This is a help for people learning German, as these verbs are either particle-prefix verbs (the separable) or phrasal verbs (the inseparable).
How to decide, which one it is?
You can distinguish between a prefix and a particle by considering whether the prefix/particle can be a word itself. A particle can; a prefix cannot. If you’re not certain, you could use some (incomplete) lists (e.g. the German Wikipedia’s one).
To distinguish between a particle-prefix and a phrasal verb, you can test the morphological separability (morphologische Trennbarkeit):
Try to build the infinitive with zu. If it is possible within the verb (like in Sie hat vor, aufzustehen), then it's a particle verb. If not (like in Sie hat vor, überzustehen), then it's a particle-prefix verb.
Or you can build the past participle (Partizip II): If the syllable ge is possible within the verb (like in aufgestanden), then it's a particle verb, else (maybe) not.
In some odd cases, this can only be a positive check, cause when it’s not possible (like in your example miteinbezogen). Then you have to rely on the other test.
If you have a good feeling for German language or have a German native speaker available, you can go by accent when pronouncing the word (or upon the context).
The University of Leipzig has a neat and short summary with lots of examples and some tests; unfortunately, in German.
Apparently, I have no English sources, as this construct doesn’t exist in the English language.