Question: Is it necessary for students to type their name on the on-line assignment that they submit, or is Moodle going to make it obvious whose work it is?
No and it depends. Moodle keeps track of EVERYTHING, so the identity of the student submitting an assignment is known. However, I usually expect students to include their name on formal assignments. They should always identify themselves on course work submissions, regardless of medium.
For discussions, I “sign” all my postings with just my initials – not really necessary, but I want students to realize it is me, the instructor, posting.
Moodle automatically displays the poster’s name and picture. I encourage students to provide a picture or an image for their profile which shows up in discussions and in lists of submitted work viewable by the instructor. Even if you don’t have a “face” to go with a posting or a quiz submission, you have a “visual” for the student.