Perhaps I could add something here too—and perhaps it will be useful: You are right that there is a difference between the two statements that you offer, and the difference has become more significant with the rise of many-valued logics in the 20th and 21st centuries.
If one says, “A is either true or false,” then there are only two possible values that A can have—true or false. But if one says, “either A or not-A is true,” then there might be all sorts of values that A could have: true, false, indeterminate, probably true, slightly true, kind of true, true in Euclidean but not Riemannian geometry, and so on. The first formulation allows only one alternative to “true” (namely, “false”), but the second formulation allows many alternatives. The second formulation does indeed require that at least A or not-A be true, but it puts no further restrictions on what other values might substitute for “true.” (For example, perhaps A is true, and yet not-A is merely indeterminate.)
The advantage of sticking to the first formulation (often called the principle of bivalence) is that it forces us to reason from propositions that describe what is definitely so or not so, and as a result, we can actually prove things. (After all, if we were to give reasons that were neither true nor false, then our reasons would seem to end up proving nothing. Imagine, for example, someone saying, “I believe this conclusion for a good reason, but my reason is neither true nor false.” Moreover, if the conclusions we wanted to prove were also to turn out to be neither true nor false, then they would remain unprovable; what would it mean, one might ask, to “prove” the untrue?) Considerations of this sort led Aristotle to believe that scientific knowledge always depended crucially on propositions of argument that had to be true or false.
On the other hand, there are many situations in life where our ideas are so vague and indefinite that the best we can say is that a particular proposition seems somewhat true, or true to a certain degree, or true for the most part. (For example, Aristotle held that propositions of ethics were sometimes only “true for the most part.” In the Middle Ages, a number of logicians wanted to use “indeterminate” as a truth value, in addition to true and false, and in the 20th and 21st centuries, logicians have experimented increasingly with the idea that there could be many truth values, in addition to true and false. As a result, there are now various systems of many-valued logic, including so-called fuzzy logic, which assigns numerical degrees of truth to different propositions.)
All the same, the principle of bivalence still plays a fundamental role even in systems of many-valued logic, albeit at a higher level. (The second formulation that you have cited is now termed the law of excluded middle, though before the development of many-valued logics, the two formulations would have amounted to the same thing.)
Specifically, many-valued logics assign different values to various propositions and then draw conclusions from the assignments. (For example, if A is “somewhat true,” then one can conclude that A is not “entirely false.”) Nevertheless, such systems always rely on at least two crucial assumptions: (1) the propositions in question, such as A, do indeed have the assigned values or they do not, and (2) these propositions cannot both have the assigned values and not have them. The first assumption is the principle of bivalence all over again, though at the “meta” level (meaning that it applies, not to A, but to statements about A, that is, to the statements of A’s truth value). And the second assumption is the traditional law of contradiction. (For more on the law of contradiction, you might see Questions 5536 and 5777.)
In other words, the propositions treated by a system of many-valued logic are typically imprecise and indefinite, and what a many-valued logic then does is allow us to talk in a precise and definite way about the imprecise and indefinite. To achieve this result, however, the system’s own statements must be definite, and to achieve coherence, the system’s own statements must also be noncontradictory. By contrast, if one were to relax these restrictions on the system, then all one would get would be an indefinite discussion of the indefinite, or an incoherent discussion. And if this last result were all that one hoped to achieve, then there would be no need to build the system in the first place. Instead, just leap from bed in the morning, and without drinking any tea or coffee, start talking. If you are like me, you will then arrive almost instantly at the appropriate level of grogginess.