Assessments are assessments. Some are more robust. Some are more formal. But at the end of the day they are meant to help us get a handle on what someone knows and can do in a particular domain. If the primary purpose of an assessment is to diagnose and provide intervention, then it is formative in nature and should not be included in making a summative judgement about the student's level of achievement (especially if there is another opportunity to demonstrate achievement). There are usually some difference in the types of assessments we design for formative and summative purposes, but they could be identical. In the end it's what you do with them that defines them as formative or summative.

Formative assessment (including what we typically refer to as homework) is really about practice. As important as practice is, it is not the same thing as the "game". Is it connected? Sure. But, it's not the same. Allen Iverson had a 29 point per game scoring average in the post season, but one issue that was reaised from time to time was his engagement during practice. While one could argue that he might have been even better if he practiced harder, his game (a.k.a. summative performances) put him in the hall of fame. 

In the second video, a skateboarder has 18 attempts to land a trick. How do you represent that in a grade book? Do you average all the attempts together?