How To Cope With The Stigma of Quitting a Second PhD for a Third one? As per the title, I quit my second Ph.D. after 10 months, having quit my first after one year. 

My first supervisor was narcissistic and a sociopath. (Nonetheless, I received prizes for my research.) After one year I left. 

I took a lab position, in an isolated country, whilst seeking a second PhD program. The lab took advantage of my situation (low salary, humiliation, harsh work conditions). After 5 months, I was about to commit suicide, until I started a second Ph.D. I thought this was the salvation.

I didn't know my second supervisor. They seemed nice. Ultimately, I took the position out of desperation. However, after several months, I found myself stuck with a supervisor who was always jealous, in a constant fighting mode, moody, angry like infant. The final straw came when I was nominated for an prestigious award, which I was to collect at a conference, which waived the usual conference fee: My supervisor refused to support my travel, even though it was only around 500 Euro. They weren't constructive at all. Perhaps more importantly, the lack of dataset from our collaborators, sketchy understanding about my research which leads to lack of feedback,  this was a wrong match in all levels.

I've found another lab in a different country. They are interested in my work and I've joined their 
group. 

My question: I feel so much stigma about the situation and I feel sometimes insecure about what happened. In my current position, they know my situation, but they did not know all the story.

Now, I have to move by the end of the month. But, I am torn. I feel a failure because this situation doesn't sound good. Even though I asked many wise people and having a discussion for hours to split the emotion and facts about my situation that my decision to leave is right, I am stuck somehow emotionally about all experience, I saw a psychoanalyst, but not so much helpful, how can I get out of this stigma, I am afraid this would jeopardize my academic career? How I can remove this past?
First things first, if you have suicidal thought, then you need help. The fact that you tried one psychoanalyst, and that did not work for you, does not mean that the entire mental healthcare system is useless for you. Finding a therapist and a therapy that works for you can be long process of trial and error. Don't give up.

As to your question: You can't remove the past, all you can do is focus on making a better future. 

If you want to reflect on your past experience to try to learn from it: I notice that you externalize the blame a lot. In the way you present your story there are three person: you are a top-student who wins prices, while the two others (the advisors) are narcissistic sociopaths or jealous and childish. That is probably not the whole story, and sticking to it will probably prevent you from learning from your experiences. However, now may not be the right time to process this. The only advise I can give you is: find a therapist that works for you and discuss this with her or him. 

If you worry about how to present your story to others, then it is this black and white presentation that raises lots of red flags with me, not the fact that this is your third attempt.