Let me just add that while you don't need to write the date received on the back of the check, you could.
Why? Let's say someone was late in paying you and you wanted to document the fact that they were late. I've had late-paying customers send me a check dated on the due date but really they just pre-dated the check and sent it 60 days past-due. So let's say I want to establish and document the pattern in case it becomes a future legal issue.
When you deposit or cash a check, an image of the front and back is made and the person or company who issued the check will have those images stored as part of their transaction history. (It used to be that the original, physical, cancelled check was returned to the payer, but that was another era.)
So write the date received on the back next to the endorsement, endorse the check, and take a photo of the front and back (along with the postmark on the envelope) to document that they are a late payer. This way, if it ever becomes a "he said she said" issue you can easily show they have a history of paying late. If the payer looks at their check images they'll see your received date note next to the endorsement.
Granted, this is a lot of trouble for a unique situation. In 20+ years of running a business I've actually had the foresight to do this a handful of times with habitual offenders, and in (only) one case did it come in handy later on. But boy was I glad to have those photos when I needed them.