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The Maria Theresa restrikes were also made in Unauthorized Mints by official government entities (e.g. Birmingham, England after 1935) - Are they counterfeit? Regular Restrikes? or in a class of their own.
I'd call them "restrikes", along with the "official restrikes" from other mints. These coins might have all the hallmarks of a "counterfeit", except for one key characteristic: intent to deceive. They were made in full public knowledge; they weren't secret or covert. The other "genuine restrikes" aren't particularly valuable, so they weren't made to fool collectors. And the locals in East Africa didn't care about "genuineness", so long as the design and silver content were the same (which they were), so they weren't trying to fool the locals either.
In fact, the only people "fooled" by these are newbie collectors who are amazed that you can buy coins from 1780 in such good condition, so cheaply.

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If an off metal planchet is smuggled into a mint struck using real dies and tehn released along with original coins - what is it?
I'd have to call it an error - unless you've got evidence of malicious intent (like, say, a rather large number of these occurring).
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If the same off metal planchet were used with the knowledge of the government but not the public does it make a difference?
Yes - those would be a perfectly genuine variety, assuming such knowledge does in fact eventually become public. If not, there will always be a cloud over them - such as with the ancient Roman plated coins known as "fourees". They're still debating over whether or not at least some of those were deliberately and officially made that way, as a secret government cost-cutting measure.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis