Difference between the Sith and the Dark Side of the Force
First of all, being Sith and being a follower of the Dark Side are two separate things. All Sith are Dark Side force-users, but the inverse is not true.
The Witches of Dathomir for example, use the Dark Side of the Force but are explicitly not Sith.
And to make it even more confusing, the Inquisitors (fallen Jedi that had survived Order 66) were trained by Vader but only considered "sith adepts" and not true Sith. One might be tempted to say, that they were Sith existing outside the Rule of Two.
It follows that falling to the Dark Side doesn't automatically mean that you turn Sith. Sith have history, goals, methods, and even rules of their own. You have to embrace those - at least to a degree - as well to become Sith. Both the Sith and the Jedi are considered religious orders after all. Becoming one is more than aligning yourself with one side of the Force or the other.
These nuances are sort of left out though, at least in the movies. The general implication is that a fallen Jedi surely would become a Sith because the teachings of the Sith promise a path to even greater power.
Side note: Dooku is actually kind of point/counter point to this
argument. He wasn't exactly a "fallen Jedi" but he had left the Order
voluntarily before he was approached by Sidious and turned towards the
Dark Side. Eventually becoming a Sith.
Why Darth Sidious provokes attacks
As to the whole "Kill me"-deal: That is meant to coax Sidious prospective apprentices and successors into following the path to the Dark Side even further. The Dark Side provides great strength and power. As such it offers great temptations. But trained Jedi especially need to be coaxed into forsaking their training. Into acting against what they have been taught and against their moral compass. They need to be coaxed into letting go of their control and opening themselves up to the Dark Side of the Force.
Emotions play a role but they are only a gateway to make people embrace the Dark Side. Often out of desperation. Sidious used Anakin's fear for Padme's life to lure him towards the Dark Side. And also encouraged him to give in to his rage and thirst for vengeance when he told him to kill Count Dooku.
It's not about actually killing Sidious anyway.
Sidious clashed two times with Yoda, the greatest Jedi Master of his age. Once physically when they dueled in the Galactic Senate and once mentally, when Sidious tried to break Yoda while he was searching for immortality in The Clone Wars episode Sacrifice.
But Yoda never managed to really best him and in fact, they are considered equally matched.
From that follows that
- Sidious can defend himself against pretty much anyone who attacks him openly if he wishes
- for anyone to have a real chance at beating Sidious, they would most likely have to embrace the Dark Side.
And that last one is what he's trying to provoke. Or at the very least for them to further give in to their emotions and thus open themselves up to the Dark Side and its possibilities.
He's actually counting on the attempt on his life to fail so that he can offer to teach the way of the Sith and finally corrupt his would-be apprentice.
So you're right in your point that Mace Windu would not have become a Sith after killing Sidious. Because he wasn't tempted by the Dark Side in the first place.
But he wouldn't have succeeded anyway. Sidious was just waiting for Anakin to intervene so that he would fall deeper towards the Dark Side. It's the same principle actually, only he didn't try to provoke an angry attack on himself but instead coaxed Anakin into betraying someone to protect his "friend".
Side note:
There is actually quite a debate whether Mace Windu defeated Sidious
or not. The official answer is apparently yes, he beat Sidious in
a light-saber duel. But Sidious was stronger in the Force and it looks
like he might have held back in order to coax Anakin into defending
him. I'm just gonna let this stand in regards to my above arguments.
At this point, Anakin had taken several steps towards the Dark Side. Specifically slaughtering a whole camp of Tusken raiders that had killed his mother (Star Wars: Episode II – Attack of the Clones) and then killing Count Dooku in cold blood (Star Wars: Episode III – Revenge of the Sith).
But betraying Windu was probably when he essentially became a Sith. It was a point of no return where he finally turned his back against the Jedi Order. And making people cross such lines is exactly what a Sith lord does to make them turn towards the Dark Side.
Sidious's reasons to provoke Luke and Rey
Now, how does this translate to Luke and Rey? In Luke's case, it was quite the same. Sidious coaxed Luke into fighting with Vader, actually hoping that Luke would beat and kill his father - much like Anakin had once killed Dooku - so that Sidious could take him as his new apprentice.
With Rey his goals were a bit different. He never wanted her as an apprentice. Instead, he wanted to take over her body. To do this he tried to coax her into embracing her inner dark side and also to lower her guard. And then he would have used his power to take her body.
