In Star Wars, can a Jedi kill?

As we see in Star Wars: Episode I - The Phantom Menace (during the climactic fight between Darth Maul, Qui-Gon Jinn and Obi-Wan Kenobi) and Star Wars: Episode II - Attack of the Clones (when dozens of Jedi storm the arena where Obi-Wan Kenobi, Anakin Skywalker and Padmé Amidala have been sentenced to death), Jedi have no qualms about killing those that are trying to kill them.

In fact, we see that Obi-Wan Kenobi is referred to as General Kenobi in Star Wars films set after the events of the Clone Wars, so we can be reasonably sure that there is nothing prohibiting Jedi killing people in battle.

However, at the point you are asking about in Star Wars: Episode III - Revenge of the Sith, Count Dooku is an unarmed opponent and poses little to no further threat. The Jedi way would have been to arrest him, not murder him, as Anakin himself correctly says in the quote in the question.

"Execute" is a subset of "kill". If it were a venn diagram, the "execute" bubble would be a tiny circle within the much larger circle of "kill".

The specific act of killing Dooku (who was, at that point, defenseless and could have easily have been taken into custody with minimal risk) was not just a killing, it was an execution. That is what was being referred to as not being the "Jedi way".

To grasp it there are modern day equivalents. A soldier on the ground in a war zone might be given a medal for killing in combat under many circumstances. That very same soldier would be tried for war crimes for walking up to a defenseless enemy that had been bested and subdued and was no longer a threat and executing him after the battle. So even to a modern soldier, the phrase "That's not the soldierly way." would absolutely apply to that situation. Yet, would you describe that as meaning it's not the soldier's way to kill? Absolutely not. It's definitely the soldier's way to kill when it's called for. Same for the Jedi.

Just an additional note from someone who read every Expanded Universe book there is before Disney's version and even Episode 1-3 came out.

Jedi are ultimately pacifists. If during a fight a sith surrenders a Jedi will stop fighting and arrest them and even better rehabilitate them back into a Jedi especially if they were a Jedi to begin with.

Jedi don't use certain force powers because of the powers destructive nature. This is the case with force lightening. Palpatine is not the only one who can channel the force to produce electric current. Some Jedi can't do force lightening because they access the force in a different way. For example some Jedi don't have telekinetic abilities. Some Jedi can only do force lightening because their species can actually manipulate energy better or actually produce electrical current on their own, such was the case with Jedi Master Arca Jeth. In general though Jedi stray away from powers that cause suffering and destruction (remember suffering leads to the dark side).

So Jedi are pacifists, and they don't use destructive powers. And now it all comes down to why. Jedi get their powers from the force. The force is literally the life energy of the universe. When Alderaan was destroyed the Jedi were weakened (as were the sith) because of all the loss of life. To kill only goes against the force and weakens one's own powers. This is why the sith are perverted. Every time the sith kill on the scale of planets they actually weaken themselves and this is why the Jedi will always be stronger - in part because jedi are accustomed to fighting without abusing the power of the force, they channel what they need (sort of like superman) they do not give into anger and they do not let the force flow erratically through them like a dam breaking. Hence, Jedi know self control - and sith do not.

Sith are temporarily stronger because the dark side allows them to channel more force at once but they are dependent on the force after, and if put in a situation were their power was drained, or alternatively where they channeled more power than they could handle at once, they wouldn't be able to compensate or adapt as well as a Jedi would. Because Sith are so dependent on the force they also burn out and shrivel up like a prune like Palpatine.

Palpatine wasn't that way from being old, he was that way from using extensive power to hide himself among other things. Which is why he started cloning himself to compensate for his failing body. More importantly Sith usually number in the one or two's and Jedi are often in the ten's to hundreds. Just a handful of Jedi together can amplify their power to throw star destroyers across the galaxy (something 1 or 2 sith could never do), or they could even maintain a temporary force barrier that protects a whole planet from an armada in space.

This is why its borderline ridiculous in the prequels that jedi council members could not maintain a force barrier against a droid army. Unless their powers were somehow being dampened by a powerful Sith Lord, then a handful of Jedi together like that is nearly unstoppable.

The Jedi philosophy during the the prequel era was about the Jedi failing because they were being manipulated by the dark side AND because they couldn't live up to their own standards...

Padmé Amidala: Are you allowed to love? I thought that was forbidden for a Jedi.

Anakin Skywalker: Attachment is forbidden. Possession is forbidden. Compassion, which I would define as unconditional love... is central to a Jedi's life. So you might say that we are encouraged to love.

It's not a matter of "if" a Jedi can kill (anyone can), but what state of mind a Jedi is in when he does.

Initially in this era, the Jedi were meant to be negotiators and keepers of the peace, but because the rise of the separatist and the reveal of a clone army, the Jedi became the Republic's soldiers working side by side with the clones, allowing characters like Obi-Wan Kenobi to be elevated to "General" status and in which they are more heavily engaged in battle and conflict.

But also, as seen in Revenge of the Sith, a Jedi is not to kill an opponent in either rage, hate, or to feel a sense of gaining power by doing so, because that is seen as a violation and corruption in moving towards the dark side, which is why Palpatine encourages Anakin to finish a disarmed Count Dooku and not bring him to justice.

As a side note, The Jedi Philosophy or applied methodology surely had some flaws, including some Jedi not owing up to their own fears about fear, and the original trilogy never really addresses the need for Force philosophy evolution only Darth Vader's redemption back into Anakin Sywalker and the alleged destruction of the Sith, but the sequel trilogy, especially the The Last Jedi, called attention back to that. And it might be possible that, like the former EU that came before, the future of Star Wars in the era post The Rise of Skywalker may be an era that evolves the Jedi philosophy forward in some way.

Jedi can and do kill, but they must do it in self-defence and or in the defence of others especially the helpless and the weak. This was allowed and encouraged by the jedi code especially when dealing with sith and dark jedi, it was accepted that killing their enemies was sometimes the only course of action to take.