I know this question cuts across quite a few centuries and places -- I'm hopping the collective power of /r/askhistorians (or one widely read historian of state executions) can answer it. Mods have pity!
Yes and no. Give me five minutes to come back with sources. There were methods of capital punishment preferred depending on which society and culture you were in but I would hesitate to describe them as national ones. For a start, that requires an idea of nationality. In certain societies, some methods of execution were unique or unusual in ither countries, but that was unlikely. Capital punishment methods were chosen to fit the society (is capital punishment purely to remove criminals from society? To scare people into being good? To humiliate and damage the body so they can't enter the afterlife? To humiliate the criminal and cause as much suffering as possible, for revenge? To be seen as making the world equal- as the victim of a criminal suffered, so would the victim?) and also the environment. You were not going to find crucifixion in a society where wood was limited, or find execution by lethal injection at a time when injections did not exist. Equally, you were not going to find one form of capital punishment reserved to one country if that country had an empire, as it was common sense to export your form of capital punishment- if you'd found what you thought to be the most convenient, efficient and moral form of execution, it would be stupid to suddenly change it. So a British form of execution- hanging- would be exported to all the colonies, and would often hang on even after colonisation had ended. Does that mean the form of execution is British- or British and the colonies'? If it continues after colonisation, despite being a remnant of the empire, I would say it was really that country's form of execution as well as British, fkr example.
Anyway, I'm getting off the point. My, uh, most specific area of study is crucifixion. It could be described as the quintessential Roman method of execution. Crucifixion, which they exported throughout the empire, was a brutal punishment. I wrote a long post on the method of crucifixion which I will need to find now. One of the key things about Roman crucifixion was that, before hand, there was a flogging, known as scourging. The victim was half dead by the end. This gives you a key insight into the mindset of those ordering capital punishment and designing the execution method- it was brutal, drawn out, public and humiliating. They wanted the victim to suffer, to be shamed in front of everyone and to be made to feel helpless and weak.
Crucifixion is a nasty way to go, and that's an understatement. But it wasn't just a Roman method of crucifixion. Crucifixion occurred in Islamic societies, in Japanese society as well. Humans are basically the same the world over. Crucifixion is a clever, horrible way to die. The victim is helpless- practically spread eagled- their injuries are in full show, it is long and allows the maximum amount of people to see it. It wasn't commercialised like hangings could be, either. Multiple people could walk past and see the victim- alive and dead. So could crucifixion be considered Roman, Islamic, Japanese? No. It's for these reasons that I don't believe there any national execution mature because every idea, someone else might have had. Execution methods are chosen for the society and all you need to do is find a similarly brutal or moral society, and you could find the same method. Of course, how that society became moral, brutal etc is a matter for debate.
To keep on with Romans, Greeks had very different methods. They essentially allowed self exile (which could be considered a death sentence for some), suicide in lieu of execution, ingestion of poison (Socrates died due to hemlock poisoning, for example, however hemlock was expensive and this was rare- suicide using that poison, rather than the state carrying out the poisoning was more likely) of a bloodless execution involving strangling. This left the body undamaged, something important for the Greek society (and for Romans too, but they still flogged victims before death, see the difference in attitudes?). It also highlighted that Athenian society was peaceful. This was not a bloody violent battle, this was Athens and these forms of capital punishment - poison and strangling- and the resultant relatively little damage to the body shows this. In contrast, the Romans encouraged that aspect, encouraged a violent and bloody death.
This isn't addressing national capital punishments, but rather why different nations would be drawn to different ones. I noticed all the examples provided are Western- but usually were not only in those Western countries. Execution within one continent may very dramatically between countries (France vs Germany vs Britain) but outside of that region, you will probably find a country with a similar outlook towards death at least, and there find a similar capital punishment, or similar style of capital punishment.
So in Rome for example, crucifixion was something very Roman to those living in and around the empire. But looking across history? Not so much. Hanging might feel very British to us but it has been used by multiple countries in various forms for generations. Shooting squads could well seem a military form of execution, and a relatively modern one, but shooting squads have been used outside of the military for civilian executions, before guns (using arrows instead), in the Roman Empire. Across the world and history, I doubt you'd find a country with a completely unique outlook on death and vapital punishment and there you won't find a completely unique style of capital punishment. Maybe a completely unique specific method, but the general style (quick, painless, medical vs bloody, long, painful, for religious vs social vs revenge purposes, etc etc) will not, the thoughts behind it will not be.
To look at very specific methods, such as the guillotine, then that's a bit closer. Your quote 'the French preferred the neat, surgical precision of the guillotine' shows the thought behind the guillotine and the type of society France was when it was used, or other countries when it was used. Modern, mechanical methods of death are more likely to have been more unique to certain countries because if the other countries have got a well established tradition, they're less likely to want it changed and adopt a new method, but even if the method is unique, the style and the reasoning behind them will not be unique.
So we can look at perhaps three different forms of capital punishment: beheading, guillotine and the electric chair.
Beheading with a 'heavy axe' can't be considered a German thing. Beheading was and even is common all around the world. The implement may vary depending on different countries' traditions, but execution by beheading existed in England (see Ann Boleyn, who was executed using a sharp sword as a concession of, well, kindness- usually they'd use a blunt, broad sword or an axe) and continues to exist even today (see Saudi Arabia). The Romans beheaded their own citizens (if they did not commit treason, that is) preferring crucifixion for non citizens. Beheading, for example, has a long tradition in the UK. As mentioned above, Ann Boleyn was beheaded, but decapitation was used as a method of execution since Anglo Saxon times. Finland practiced beheading with an axe as did Norway and Sweden, despite very different cultures, with the last Finnish beheading being in 1825. Other countries use beheading but with a sword or a sharpened knife, not an axe, including old New World societies, who developed their form of capital punishment without influence from the Old World.
Guillotine: the guillotine was exported throughout the French empire, but very similar methods occurred in other countries too. The Halifax Gibbet is a prime example, predating the invention of the the guillotine. It was considered a more humane alternative to execution by beheading, and it appears the Halifax Gibbet only existed in Halifax (Yorkshire, UK) despite decapitation being a common form of punishment throughout the country, and it was used from at least the 16th century. It was slightly different and was bigger than the guillotine. Another guillotine like device was the Maiden, used since the late 16th century to the 18th century, in Scotland. Supposedly the inventor borrowed the design from the Halifax Gibbet, although we can't know the truth. In Germany, the guillotine also existed, called the Fallbeil or falling axe, dating from the 17th century. Up until the French Revolution, I would argue that guillotine like methods were a Germanic system of execution rather than anything else. The thoughts and reasoning behind the mechanisms were similar across those societies and also in other societies who beheaded people. The idea behind the guillotine wasn't unique and neither was the specific method or style.
The electric chair was to be a more humane alternative to hanging (and that sentiment was common, the idea behind the electric chair therefore isn't unique). It was only really used in the US and the Philippines (spread by the US). That could be considered a national punishment because the cost, the time period ( when execution was falling out of favour) and the efficiency made it unattractive for other countries to adopt it. Electrocution could be considered an American method of execution, although not exactly a purely American style- many countries have tried or have used modern mechanisms and discoveries to provide a supposedly more humane method of death, in societies with similar views towards death. How the electric chair is set up, how the executor is removed from the execution, how the sentiment behind it was to be moral and ethical, that is quite common.
I was inspired to ask after seeing the week’s theme and recalling this passage in David Anderson’s book on Mau Mau, Histories of the Hanged:
“The Germans used to behead the convicts with a heavy axe, while the French preferred the neat, surgical precision of the guillotine. More recently, the American lust for artful technology, even in death, has drawn them toward the electric chair and the lethal injection. In Britain, the execution of choice has long been the rope, the nose, and the drop. The British public has always liked a good hanging.”