Now, let me explain all the pieces here, as that went by a little quickly, even by my standards. The first bit is the link between reasoning well and reaching the goals of seeking truth and avoiding error. I think that follows from the discussion above. There are forms of reasoning, scientific reasoning being the clearest case, which allow us to seek truth better than otherwise. So, if you're good at scientific reasoning, for example, then you're going to be good at seeking truth. And being good at reasoning is the same as reasoning well. The same works for the syllogistic form of reasoning which keeps us from falling into error. If you're better at constructing syllogisms, then you're better at reaching the goal of syllogisms, namely avoiding error. And being good at reasoning is still the same as reasoning well.
The second bit is the link between the two goals and thinking better. Why is it that seeking truth or avoiding error is what constitutes being a good thinker? There are other things we might consider important to thinking well -- clarity, for example, or creativity. But I don't see how other cognitive goals could be really worth having unless they served to either seek truth, avoid error, or both. Take clarity, for example. Someone thinks more clearly when their thoughts are easier to understand, can be expressed more straightforwardly, aren't filled with confusion or vagueness, and so on. Now, in what circumstances is it actually valuable to think clearly rather than obscurely? I find it hard to think of any, except those where clearer thoughts are thoughts that get us closer to truths; or, alternately, where clearer thoughts are thoughts that help us to see mistakes and avoid them.
Consider the contrary case: where someone is a very clear thinker, but is constantly falling into error and believing things that are false. Is it really such a good thing that this person thinks clearly? Is clarity any longer a feature of thinking well? I don't see how it could be.
Similar things can be said for creativity. Obviously, there are very creative thinkers -- many work in politics, many more in movies, even more in religious institutions -- who think in ways which lead them into believing false things and into making mistakes. Some of these false things are benign, even entertaining, as in the case of creative writers who come up with fictions (fiction, after all, being strictly a form of lying); and some of these false things are very dangerous, as when a political leader starts to believe he can cut his country back to prosperity, or a religious leader believes that his deity commands him to fleece his flock for as much as he can. So, creativity, like clarity, is far from a bad thing. But when it isn't attached to seeking the truth or avoiding error, it seems to me, again, that it's really not a valuable thing.
The tricky case is the case of fiction, as we do tend to believe that fiction is valuable, and it seems to require creative thinking which gets us away from what is actually true. However, I think this can be misleading. Fictions aren't wholly false, after all; even a fiction very far removed from our own experience -- say a story of a man battling monsters after being transported to Mars -- is still grounded in something familiar and true -- in this case, the man, his battles, the planet Mars. Furthermore, the best fiction provides true and useful insights into real things; the fictional setting is used as a device to expose us to truths about what people are like -- think of Golding's Lord of the Flies or Orwell's Animal Farm -- or expose errors in our ways of thinking about the world -- historical fiction is often surprising in this respect. [Note: Of course, historical fiction isn't always fully accurate. But, I maintain, the best and most memorable historical fiction makes the past come alive by showing us what it was really like, even when that conflicts with what we would like it to have been.]
Finally, there's the link between reasoning well and logic. It seems to me that if logic is anything at all -- if it isn't just an empty name which attaches to nothing, like "round square" or "the present King of France" -- then it has to be a matter of reasoning well. As said, this seems to me to be just definitional. If I were being technical, I might call this a "platitude" or "conceptual truth". But these are bombastic ways academic philosophers have conjured up to say that there really isn't any other way to conceive of the one idea without conceiving of the other. So, it's a "conceptual truth" that squares are rectangles. The very idea of a square contains within it the idea of a rectangle; it's impossible to bring to mind the concept of a square and not have brought to mind, possibly inadvertently, possibly without realizing it, the concept of a rectangle. Similarly, it's a "conceptual truth" that all effects have causes. The very idea of an effect contains within it the idea of having been caused. So, in the same way, the idea of logic brings with it the idea of reasoning well; and reasoning well brings with it the idea of logic.
That's the argument, then. But this is all very abstract. How about some examples of how this thinking well by reasoning well works? That is, cases where improving our reasoning ability helps us to achieve the two goals, and thus count as cases of improved thinking -- that is, cases where logic is helping us in just the way I said it should.