How to explain past PhD withdrawal to employers/potential supervisors?
I wouldn't go into a lot of detail.  The person asking you this wants one single answer, and it should be something that puts you in a good light for the new opportunity and doesn't provoke hard follow-up questions.

If you want to go into the workplace, I would say &quot;My previous program focused too much on theory without useful application and I really wanted something more applied.  I like to work on projects that have useful application in the real world.&quot;
Alternatively, if you want to be in industry, you can say that you realized you actually don't want to work in academia and your previous program was interested in keeping you in that realm, rather than preparing you for the workplace.
If you are pursuing more education, you may focus on differences between the field you quit and your new one.  Something like: &quot;I realized I'm really much more passionate about computer science than I am about pure mathematics.&quot;
You should do nothing to mention the issue in anything that you give to everyone.  That's information to be conveyed in an interview, if at all.

It's a given that there were probably other things going on and your interviewer will assume there is more to it, but they are not your therapist and do not need to understand the whole situation.  Only give them what is sufficient and positive.
I was kicked out of a Ph.D. program at a very late stage because I simply was not performing well.  The main reason for my poor work was personal issues that are embarrassing and were my fault but were not directly related to my studies.  Later I re-applied to a program in the same field. When they asked why I dropped out I said that my advisor was travelling my whole last year and unable to supervise me well and that I had a compelling offer in industry. Both of those are true and were significant factors in my outcome. I didn't tell them I was also having personal problems at the time that negatively impacted my ability to focus on my work.
I got into the program and finished my Ph.D.  On the job market people again asked me why I did a few years at one school before finishing at another.  I said various things similar to the above but was never in a position where I really had to explain it.  I did some studies at one school, went into industry for a while, and then went back and finished at a different school.  That's as much as people know.  To this day my colleagues don't know what the deal was and have never had enough interest to find out.  That's the way it should be---professional conversations should be kept professional and private information rarely has a place in them.
I have seen people play the &quot;I was young and immature&quot; card and seen it work out, but it's a big risk.  My advice is to be honest with people, but don't undermine your future by dredging up potentially harmful issues from your past.  Instead say things that are true, put you in a good light, and will satisfy the asker.