Whether/how to list a failed student on my CV
One thing to keep in mind is that tenure-track faculty generally do not list failed PhD students on their CVs.  One might argue that they should actually, since -- unlike a non-academic resume -- a CV is supposed to be a factual, comprehensive record of one's academic career, but nevertheless academic culture doesn't cut this way.

The next thing I will say is that in my experience at least, "co-advisor" refers to a member of a group of more than one faculty member -- usually, though not absolutely always, tenure-track faculty members -- who jointly share the role of advising.  You seem to be using the term as a kind of "lieutenant advisor," but that could be misunderstood.  When the student's other advisor is your PI, it is likely that your advising duties are less official and more subsidiary.  In particular, you write "[M]y professor decided to dismiss him from the doctoral programme and from employment," not we decided...Nor did you hire the student, fund him and so forth.  However, from the sound of it, you played a large intellectual role in supervising the student.

So: I would not recommend you put a line on your cv which says "co-supervised a PhD student."  If potential employers are interested in that, they are likely to ask you about it, and when they find out that the student didn't graduate, I think that sounds bad...in particular, it sounds not completely forthcoming.  On the other hand, "co-supervised a PhD student who did not finish [or however you want to put it]" also sounds bad, as I mentioned above.

I would suggest instead that you put a line on your CV saying that you mentored a PhD student and that mentoring relationship resulted in a joint paper.  That's a successful mentoring relationship.  The student was not ultimately successful in the PhD program, but I claim that you're not on the hook for that: you didn't hire him and you didn't fire him.  If they ask for more information about the PhD student, you should of course be completely forthcoming about the fact that he was dismissed after six months, as well as all the help you gave him.  If/when this additional information comes to light, it does not undermine what you said before in any way.  Since you are not claiming the PhD student as your own, his fate is not your responsibility (which is true!), and so your involvement in the situation looks (and was, based on your description) entirely positive.