***H: "As far as we know, truth may only be able to be roughly approximated." The question is, approximated by who?***
Truth is an ill-defined human concept that words, equations, definitions, algorithms, etc are often used to describe the way reality is. This process of describing the way reality is, may happen to be impossible (i.e., there is no algorithm that exists which perfectly describes the way reality is - there are always exceptions that exist to those descriptions).
***Truth itself is definite.***
How do you know truth is definite? If it turned out that reality cannot be perfectly described, then how can it be definite (i.e., there exists no algorithm which can perfectly describe reality).
***What, who; exists IS ALL that, who EXISTS. Truth is all there is.***
We should clarify what we mean by truth and what we mean by reality. Reality is 'what there is'. Truth, on the other hand, is not so well defined. However, most people think of something as being true if a true/false statement about reality happens to perfectly match reality. For example, to use the widely accepted Tarski definition of truth, we can say it is true that "snow is white" if snow is white. We are comparing a statement to reality, and if the statement extensionally equates to the way reality is, then the statement is 'true'. Truth is limited to human meaning(e.g., statements, sentences, propositions, equations, etc) but truth should not be confused for the term 'reality'. Hence, reality is all there is, and truth (according to Tarski anyway - which is not universally accepted, btw)is that which is extensionally equivalent to reality in the form of a meaningful statement about reality.
***However, as far as Tarski's theory of truth goes; 'does a proposition correspond with the facts?'; that presents approximation difficulties if information is incomplete. Incomplete data may lead to mis-matching patterns through mistaken guesses.***
I agree that this is one problem who wish to use Tarski's definition to justify calling something true, but it is much more than that. If there is no way to match a statement to reality, then the meaning of the statement is a human approximation. This might be the 'way reality is'. That is, reality may not be definable and there's good reason to suspect that it is not.
Warm regards, Harv