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There are names in German for separable prefixes (trennbare Präfixe) and inseparable prefixes (nicht trennbare Präfixe). Prefixes that can be either separable or inseparable are descriptively called Präfixe, die sowohl trennbar als auch nicht trennbar sind.
Is there a term in German to describe a two-component separable prefix? One example of a verb with such a prefix is:
miteinbeziehen |bezog mit ein / miteinbezogen|
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).
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).
This is a help for people learning German, as these verbs are either particle-prefix verbs (the separable) or phrasal verbs (the inseparable).
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.
As far as I am aware there is no such thing as you are describing. @biolauri suggested the term Doppelpartikelverb, but it is actually a term originating from the field of Morphologie, which isn't applicable.
Partikelverb: syntactically seperable; verb elements for a verbal "bracket", where the left sentence bracket is occupied by the finite verb stem and the right sentence bracket is occupied by the particle. zu in the infinitive form is word-internal, as is ge- in participle form.
Examples:
umgefahren, "Er fährt ihn um."
heimgehen, "Er geht heim."
This looks like what your supposed "two-component seperable prefix" indicates.
Doppelpartikelverb: Behaves like a Partikelverb, but has a "double particle" (not two particles).
Examples:
hineingehen (hin+ein+gehen), "Er geht in das Geschäft hinein."
hinterherfahren (hinter+her+fahren), "Wir fahren ihr hinterher."
On first glance you think this would be a representation of what you're looking for, but in (nearly) all cases, this double-particle is non-seperable. In fact, your mentioned example formed with mit is the only example I have ever seen, where the double-particle (I am not even sure that it is or isn't a double-particle) is seperated.
It very well may be, that when a particle is formed with "mit" there is a special rule that the seperated particle is seperated between the two components. I can not find a rule for this, but it's a very specific case, so that may not mean anything.
All in all I have to agree with @Janka in that the category you envision isn't defined and there is likely only an implicit rule for it. Additionally, his/her description of the difference between inseperable prefixes and particles is sound.
An additional comment regarding the morphological seperability check that @biolauri suggests:
Don't try to check for the applicability of ge- in the past participle. Rather check for strong, mixed or weak verb, (check how often the stem vowel changes), which is a better indicator for morphological seperability, albeit also not without exception.
The categories are about the prefixes, not about the words having those prefixes.
The difference between inseparable prefixes and separable ones is fundamental. Inseparable prefixes like be-, ent-, ver- and zer- aren't words —not even particles—, while separable are.
There are even mixed words, e.g.
(sich) vorbereiten → Sie bereitete sich vor.
vorverlegen → Man verlegte die Prüfung vor.
So, no, there isn't a special category needed for words with more than one prefix.
Note there is a small category of inseparable prefixes which are words. Most prominent examples are über und unter.
überweisen → Die Ärztin überwies ihn ins Krankenhaus.
unterweisen → Sie unterwies ihn.
unternehmen → Er unternahm etwas.
unterstellen → Sie unterstellte ihm eine Affäre.
BUT
unterstellen → Er stellte sein Fahrrad unter.
unter can be both, separable and inseparable. Note the stress for the inseparable prefixed word is different from the separable prefixed one.
