Is it legal to call a C compiler written in C or a PHP interpreter written in PHP metacircular? Is this definition valid only for languages of a specific type, like Lisp? In short, what are the conditions that an interpreter should satisfy for being called Metacircular?
A metacircular interpreter is an interpreter written in a (possibly more basic) implementation of the same language. This is usually done to experiment with adding new features to a language, or creating a different dialect.
The reason this process is associated with Lisp is because of the highly lucid paper "The Art of the Interpreter", which shows several metacircular interpreters based on Scheme. (The paper is the kernel for the book SICP, and its fourth chapter works through others that create e.g. a lazily-evaluated Scheme.)
This is also vastly easier to do in a "homoiconic" language (a language whose code can be manipulated as data at runtime), such as Lisp, Prolog, and Forth.
As to your direct question - the C compiler wouldn't be an interpreter at all. A compiler written in its own language is 'self-hosting', which is a similar property, but more related to bootstrapping. A PHP interpreter in PHP probably wouldn't count, since you would likely be re-implementing a nontrivial amount of the language in the process. The major benefit of a conventional metacircular interpreter is that doing so isn't necessary - you can plug in the existing parser, garbage collection (if any), etc., and just write a top-level evaluator with different semantics. In Scheme or Prolog, it's often less than a page of code.