You linked Article 3 of the third Geneva Convention; all that says is that in a war not of an international character (meaning a non-state war) then signatory states to the GC must not:
- Murder, mutilate or torture
 
- Take hostages
 
- Commit outrages upon personal dignity
 
- Commit extra-judicial executions or carry out extra-judicial criminal sentences
 
Those are things that they specifically cannot do, no matter the character or nature of the war. The third GC as relates to the treatment of POWs is vast, and while Article 3 says you should adhere to all the Articles in a non-traditional war, it is not a treaty obligation.
Aside from those four prohibitions, it also puts one “required action” on the signatory power, namely they must take care and render care to the wounded and sick.
There is no prohibition against detention of persons, and the GC is not like, say, American constitutional law. There aren’t “presumptions” aside from the presumptions made about the traditional conduct of wars. What that means is, if it isn’t specifically prohibited by the GC or specifically required, then it is just sort of assumed states will behave as states always have in armed conflicts.
If you actually read through the GC you will see that nothing in it contradicts exactly what I’ve said, namely that:
- States have always been able to hold unlawful combatants. This is a 21st century term but the terms spy, privateer, or saboteur have been in the language for centuries. The practice of holding these people prisoner when they are captured is as old as the concept of privateers, spies, and saboteurs. Further, it is a concept as old as those types that they generally face punishment for their actions. In many wars in more “barbaric” times, it was of course the order of the day that you didn’t really take prisoners (if you did, it was only members of the wealthy class, and only for the sake of getting ransom out of their families.) In somewhat less barbaric times (say, 18th and 19th centuries) you would sometimes have more humane treatment for regular soldiers–however even then spies, saboteurs, and privateers were routinely hanged or executed by some other means. What I’m saying here is that is a traditional and customary practice in war. Long before formalized international law, some powers started to see the enlightenment values of releasing captured front-line soldiers after hostilities. They also saw the value in not killing these people simply because of the fact that they were fighting in a war. However, the view on privateers, saboteurs, and spies did not change.
 
During the American Revolutionary War, spies were hanged (Major John Andre, Nathan Hale), but regular soldiers were usually returned home (see: the British captured at Saratoga, Yorktown, the Germans captured at Trenton.) While some of them may have died while in captivity, it was generally the goal that those persons would be released at the end of the fighting.
The American Civil War followed a similar formula.
- Executing, say, privateers or saboteurs is not out of line with the Geneva Convention. This is because they are not POWs. The Geneva Convention is very specific that if someone is a POW they are not to be tried and punished for their martial actions. The GC does not provide these protections to privateers, saboteurs, or spies. This is very obvious because of how explicitly the GC defines who it protects. If it intended to protect everyone engaged in armed conflict, then Article 4, which defines POWs, would not have a list with something like four conditions for defining POWs explicitly. If the GC was intended to protect all captured persons with POW status, do you not believe it would just say that? Instead, it explicitly defines POW and gives protections to POWs. Then it also says, in Article 3, what the minimum requirements of signatories are, in non-traditional conflicts. Nothing in the relevant articles precludes holding persons who fall outside the realm of POWs indefinitely, meaning it’s perfectly kosher under the GC.
 
The whole structure of the GC clearly accepts and in fact establishes the concept of there being combatants who are to be afforded the specific rights of POWs, and combatants who are not afforded those rights but are given some protections. One of those protections is not “protection from detention.”
Where Bush and the United States went off the rails is the traditional and customary way to handle unlawful combatants (saboteurs, spies, and privateers) was to either a) try them and hang them or b) release them when the war was over. If you think in the contexts of history, prior to the “War on Terror” it would not have been in the interest of state powers to hold people indefinitely. What is there to gain from that? No, instead if you had combatants who had you felt were criminal, you usually had them executed (but you did give them a form of criminal trial–Andre and Nathan Hale were tried, and that was over 200 years ago.) If you were holding a large number of such combatants, maybe you just released them when the war was over if you didn’t really feel they were that egregious. This was probably more common for Privateers than for saboteurs and spies, mainly because granting of letters of marque was very common. However, that was a practice of the 17th and 18th century so the treatment of those persons was not held to any real international standards. Some states would recognize captured privateers as state agents and may eventually release them, many others simply had them hanged as pirates.
What was controversial, and I maintain this, was not holding people captured during war time as “something other than POWs.” It was holding them “indefinitely”, it has always been understand that even POWs would be held indefinitely in terms of “time.” Meaning, it was always understood a POW would be held til the end of the war, but how long that war would be of course could not be known during the fighting itself. When prisoners first started being taken to Guantanamo Bay, we didn’t even know if the government would release them after the end of hostilities in Afghanistan (or anywhere else.) They essentially existed in limbo, and that was out of sync with history. It wasn’t uncommon at all to hold unlawful combatants and then eventually release them. Nor was it uncommon to hold them and eventually try them and execute them.
Compare it to members of Iraq’s military we captured during the invasion of Iraq, they were held as POWs and it was understood they would eventually and definitely be released (aside from the ones who were being prosecuted for various war crimes/crimes against humanity committed before or during the conflict.)