Interpret the discussion of knowledge as true belief in Plato's Theaetetus, what are the puzzles for this position and precisely would there be a better way to confront them? Furthermore, why does Socrates ultimately reject even the modified version of this position, which has it that knowledge is true belief and an account?

Interpret the discussion of knowledge as true belief in Plato's Theaetetus, what are the puzzles for this position and precisely would there be a better way to confront them?
In Theaetetus, Socrates and Theaetetus discuss what knowledge constitutes. According to Theaetetus, knowledge is either one of four things: a) it is wisdom contained in the arts and sciences b) it is what our perception tells us c) it is true (belief) judgment (as opposed to false judgment), or d) it is true (belief) judgment accompanied by a logical explanation (account). 
Based on your question, you would like a discussion of c) knowledge as true belief or judgment. Theaetetus proposes that, in order to have true belief (or judgment), one must be able to distinguish it from false judgment. The puzzles for this position are as follows:
1) One problem for this position is the puzzle of misidentification, which Theaetetus equates to mistaking A for B. Socrates argues that one cannot make such a false judgment, however. If one does not know A, then one cannot know B either (and therefore cannot make a false judgement). If, however, one knows A, one will likely understand that A cannot be B.
2) Another problem is the puzzle of believing what is not. Socrates argues that one cannot equate false judgment (belief) to faith in nothing. It is technically absurd.
3) A third problem is the puzzle of Allodoxia, which equates false belief with the practice of inadvertently exchanging one piece of information for another. Socrates argues against this, as he maintains that all our thoughts are concrete and accessible to us. If we are conscious of all the objects of our thoughts, there can be no room for inadvertency. 
4) A corresponding problem rests in the puzzle of the Wax Tablet, where true belief (judgment) is acquired by what is imprinted on one's memory. According to Theaetetus, false judgment (belief) occurs when we wrongly match a memory to a perception, such as when we mistake one person for another, based on what we think we remember.
However, Theaetetus' theory does not take into account false beliefs about mathematical equations. Socrates argues that a person may believe that 7+5=11, when the truth is that 7+5=12. According to the Wax Tablet idea, false beliefs based on wrongly connecting a memory to a perception are possible, but false beliefs regarding mathematical equations cannot happen. Socrates argues that the Wax Tablet idea leaves much to be desired.
5) A final problem rests in the puzzle of the Aviary, where the birds in an aviary represent pieces of knowledge. If one catches a bird, one essentially obtains knowledge. It follows from this argument that if one catches the "wrong" bird, then one will obtain ignorance. Although Theaetetus argues that we can define false belief this way, Socrates argues otherwise. He presents a quandary for Theaetetus: what if a person obtains "ignorance" but thinks of it as wisdom or knowledge instead?
Furthermore, why does Socrates ultimately reject even the modified version of this position, which has it that knowledge is true belief and an account?
After arguing the above, Theaetetus presents the idea of knowledge as true belief (judgement) with an account (or explanation). Socrates rejects this argument because he contends that an explanation is already a part of true judgment. Therefore, it cannot in itself define knowledge. What Socrates is saying is this: to define knowledge, we still have to separate true belief from false belief; since we have yet to do this satisfactorily, how can we know what knowledge constitutes? 
https://plato.stanford.edu/entries/plato-theaetetus/