I could be wrong here, I'm not really familiar with Taoist thinking, but the way I interpreted the quote above is that it is more about being master of oneself, i.e. in control of your emotions and reactions, rather than being a master of the subject in the eyes of others. It seems to me that the point is not to care about reputation or winning the argument because you have no control over others anyway.
The first part of this chapter is important for understanding the whole, I think (especially with this translation):
"Whoever relies on the Tao in governing men
doesn't try to force issues
or defeat enemies by force of arms.
For every force there is a counterforce.
Violence, even well intentioned,
always rebounds upon oneself."
What this is saying (again IMHO ;-) ) is that in the conversation, you can't control both sides: only your side. You can't control how someone receives your words, or how they interpret them. If you try to press your case, you will cause things to occur which you may not anticipate and for which you have no control.
Therefore, if you are trying to assert some idea, rather than to clarify your own understanding, then you may actually create the antagonistic forces which you were originally trying subdue (cut off one head of the hydra and be rewarded with 2 growing).
This is one of the core principles in the tao te ching. The word tall has no meaning unless you compare it with short. If I draw a box and ask you if the box is tall, you can't really say. However, if I draw another box of a different size next to it, it is easy to distinguish which one is tall and which one is short. Similarly, if I draw a single box and call it "tall", you can get an idea of what "short" means -- definitely shorter than the box I drew.
In than way, declaration of one thing "creates" its opposite. If you make a declaration on the internet, it can create its own opposition. People who would never have thought about the issue, may come to defend the opposite point of view. The more you push your point, the more vigorous the defense. Had you done nothing, then nothing would have been the result.
In taoist literature, the middle point between 2 extremes is called the "pivot" point. There is a point between being "tall" and being "short". If I grow my short box, it will eventually cease to be short and start becoming tall (and in comparison, the other box will become short). That point is the "pivot" point.
This is a concept of "utility". Normally for something to be "useful" you must transition between an extreme and the pivot point. If you have a cup that is always full of water, then it is not "useful" (in the ordinary sense of the use of a cup -- it might still be useful as a weight ;-) ). If the cup is always empty, then it is equally not useful. It only has utility when the cup transitions between being empty and full. The same is true of spokes on a bicycle wheel. Which is more important: the wires that provide tension on the outside of the wheel, or the spaces between the wires? Without the spaces, there would be no wires: only a disk. And then there would be no spokes. It's important to alternate between spoke and space.
So, to answer the parent's question: from (my interpretation of) a taoist point of view, you need to be careful not to press your case because you will create your own opposition. However, if you wish to have utility in your argument, it is important to move from the pivot point, to an extreme and back to the pivot point again (possibly over and over and over again). How you do that is beyond my ability to answer ;-)
