So far, I have consistently condensed language, and in some cases eliminated little used appendages in presenting the Constitution. I have felt comfortable with that having given you access to the original. For Article III, Section 2, I have felt it necessary to expand on the language, since it assumes a body of structure for the courts that to many of us may be totally foreign.
That said, the concept of a court in the English tradition, starts with a body of law. The court is there to judge the actual performance against that required by the law. In Equity, there may not be any actual WRITTEN law, but there is most certainly a judge memorized body of tradition that everyone involved takes great efforts to adhere to. So the concept of a court starts with a body of law. The second assumption is that a court is a tryer of fact, which means they will hear evidence and weigh it against the credibility of the witness and relative weight of any conflicting evidence. It is the jury's job to come up with a hypothetical statement of what actually happened, beyond a reasonable doubt. If they cannot get beyond the doubt, then the case is dismissed. If the defendant is found guilty, then it is back to the judge to decide on the penalty, again according to the law as he has been given it.
When the founders wrote, "The judicial Power of the United States shall be vested in one supreme Court, and in such inferior Courts, , ,", I have absolutely no doubt that they thought they had just written out all of the above and more. There is going to be a court. Everyone knows what a court is. It's been around for hundreds of years.
So, in my mind, what Section 2 of Article III says is, "There is going to be a court. It is going to work according to all of the above. It will be multi-layer, and we are just leaving it to Congress to figure out the details on that.
If you can get your head around that, the main questions before any court in the English tradition are: What's my body of law? and Who am I supposed to apply it to? If I have a well oiled court, I should be able to take it anywhere, and given the answers to those two questions it should work just fine.
Well, in this case, the body of law is just emerging. That may seem like a problem, but in the English system the body of law is constantly in a process of emerging. As long as we know where to look to find it we are good. The answer we have been given in the Constitution is actually in the form of what cases will we hear. We are left to infer from that who it is that we are judging. Because Section 2 is constructed in such a specific way to answer questions that were not even obvious to us, I have taken the liberty of just rewriting it in a way I can follow. I hope it is helpful.