The easiest way to show this is by creating any activity with a required completion grade and failing it. I would expect that the timestarted is (at least) recorded when they fail, although it should be recorded when they enrol as the "start on enrolment" option suggests. Reproduce as follows...

• Create a course with completion on and completion tracking set to start on enrolement
• add an activity with a passing grade in the grade book e.g. a simple quiz
• enrol a user
• Run the completion cron
• check the values in the mdl_course_completions table for that user
• will have a timeenroled value but nothing for timestarted or timecompleted
• go through the activity as that user and fail it
• re-run the completion cron script
• check the mdl_course_completions table - no change
• go through the activity as that user and pass it
• re-run the completion cron script
• check the mdl_course_completions table - timestarted and timecompleted will now be set to the same value.