Neither Coq nor Isabelle are based on Martin-Löf Type Theory (in a narrow sense).
Coq is based on a variant of the calculus of construction (with (co)inductive definitions) and Isabelle is an LCF-style prover framework that you can instantiate with various different logics. The most widely used variant is Isabelle/HOL which implements a variant of Church's theory of types. To implement other logics, Isabelle uses intuitionistic second order logic based on lambda calculus, without numbers or recursive data types. Idris has Type:Type, so is unsound as a logic. I'm not sure exactly what Epigram uses, but I assume it's somehow based on Luo's ECC. Only Agda is based on Martin-Löf Type Theory.
Yeah... This is correct as far as I know though a bit out of my complete technical grasp. To my understanding MLTT was a fairly central topic and it helps to distinguish MLTT from just type theory since plain-old-TT might be harder to distinguish as something interesting above `(int *)`.
But, to solidify, mafribe is much more correct than my comment and there's, as far as I'm aware, a much larger menagerie of other type theories. The recent Homotopy Type Theory publication—which was on HN as notable due to its publication process—was yet another step in this direction.
Per Martin-Löf was instrumental in extending type theory to dependent types, although the first version of his dependent type theory was impredicative with Type:Type and was shown to be inconsistent by J.-Y. Girard. Subsequent versions of Martin-Löf type theory have been predicative, but other dependent type theories (such as the calculus of constructions) stuck with (more benign forms of) impredicativity.
