The most coherent definition I can come up with is that homoiconicity describes the relationship between language syntax and a data representation, such that all valid programs can be parsed by a conforming data parser, and the parser output contains all the information required to compile or execute the program.
Thus, all programs are homoiconic to some data format (if only flat text files). The interesting thing about Lisp is that it's homoiconic to s-expressions, which it also provides powerful tools to manipulate. This paves the way for its macro system which provides compile-time support for processing s-expressions and treating the output as input to the compiler.
Your definition (already rather loose) covers only one direction. The point of homoiconicity is that code and data are the same, so they must be easily interchangeable both ways, and also during runtime.
Even if you would consider flat text as an acceptable data format for homoiconicity, most languages don't fit that criteria. Can a random program in a random language (not expressly written as a quine) easily obtain itself in this representation? Can I easily write a function that, when passed any arbitrary function as an argument, obtain the flat text representation of that function, modify it, and execute that modified function?
I think the word syntax is ambiguous. Is homoiconicity about concrete syntax? Abstract syntax? Both? Some other kind of syntax? Does parsing have something to do with homoiconicity?
Thus, all programs are homoiconic to some data format (if only flat text files). The interesting thing about Lisp is that it's homoiconic to s-expressions, which it also provides powerful tools to manipulate. This paves the way for its macro system which provides compile-time support for processing s-expressions and treating the output as input to the compiler.