I am wondering whether I should include students I have supervised but have quit in my CV, especially for grant proposals where supervision history is assessed. I have a couple of students who could not finish their programme, either because they didn't meet the academic requirements of our university or because of lack of funds. Do you think it would look bad on my CV to include them? If I do include them, should I explain why they quit?
For grant proposals in particular, this information can be used to assess conflict of interest. In fact, the National Science Foundation (NSF) moved that section off the biosketch to a separate document to make the intent of this section more clear. There, you could also indicate the last time you interacted, which may make the lack of relationship clear.
With grants, it is always essential to ascertain the purpose of the information to understand what information to present and how it should be presented. If you are unsure, you can always email the sponsor's help desk. They are typically fantastic in helping you understand the documents you need to assemble. Do not worry that they will assess your questions with your application. PIs frequently worry about that, but that isn't happening. They want good applications that are easy to review, so ask away!
I think that would depend on how many there are, compared to the successful ones. It would also depend on the reasons and if they would seem reasonable/common to a reader.
Certainly having students quit, for whatever reason, isn't a boost on your CV. But if your CV has a section on "Successful students that I've supervised" then it would be natural to include only those who succeed.
Certainly students quit for reasons outside the control of their supervisor. People will understand that. However, if too many of your students quit, the reasons given may not be the true reasons.
I would probably include them all if they spent a year with you (post bachelors; less is fine for undergrads and postdocs and visiting scholars and the like). It is normal in some fields for there to be some attrition (e.g. Ph.D. student getting frustrated and bailing with a masters). They still did did work with you (just like an undergrad did for instance). Perhaps you don't have to really dwell on the issue of if they got done or not. Just list them in sections, undergrad, post bachelors, masters (could include Ph.D.s that bailed), Ph.D. and postdocs.