"No, that's not a belief, that's a definition of what it means to say 'I believe X'."
That's not a definition, it's an act of linguistic warfare.
If you were actually defining a word, you could replace the word by any made up string of letters, and the definition would have the same effect: an indication of what you will mean when you later use that word. "That's a definition of what it means to say 'I flamjink X'." does exactly the same work as "That's a definition of what it means to say 'I believe X'." Uttering the latter sentence tells nobody anything about what anyone else means when they use the word "believe". It merely informs them of a personal decision that you have taken about how you will henceforth use the word.
But that is not the intended function of "No, that's not a belief, that's a definition of what it means to say 'I believe X'". It's an attempt at expropriating the word, an attempt to decree not merely how other people must henceforth use the word, but to decree that that is what they always meant by it -- what the word itself "really means". If that's what it "really means", then that's what everyone previously using it must really have meant -- by definition!
Rephrasing your next paragraph:
A definition is not true or false, it is useful or not useful. Why is this definition useful? Because it allows us to distinguish between two classes of declarative statements; the ones that are actual flamjinks, and the ones that have the grammatical form of flamjinks but are empty of meaningful flamjink-content.
Doesn't sound so convincing. It may very well be helpful to distinguish beliefs that are testable from beliefs that are not, and to make up a name for the former class (although "testable" seems to serve well enough already). But none of that deprives the non-testable beliefs of meaningful belief-content.
Very brief recap: The logical positivists said "All truths are experimentally testable". Their critics responded: "If that's true, how did you experimentally test it? And if it's not true, who cares?" Which is a fair criticism. Logical positivism pretty much collapsed as a philosophical position. But it seems to me that a very slight rephrasing might have saved it: "All _beliefs_ are experimentally testable". For if the critic makes the same adjustment, asking "Is that a belief, and if so -" you can interrupt him and say, "No, that's not a belief, that's a definition of what it means to say 'I believe X'."
A definition is not true or false, it is useful or not useful. Why is this definition useful? Because it allows us to distinguish between two classes of declarative statements; the ones that are actual beliefs, and the ones that have the grammatical form of beliefs but are empty of meaningful belief-content.
It seems to me, then, that both the positivists and their critics fell into the trap of confusing 'belief' and 'truth', and that carefully making this distinction might have saved positivism from considerable undeserved mockery.