Yes, a cease and desist is just a letter with no legal obligation whatsoever. I could send one to Rockstar today. The interesting part is the threat of legal action, so I wouldn't send one unless I had a solid case. The letter itself isn't interesting, the judge's order during or after the ensuing legal battle is.
And yes, a DMCA should only apply to copyright infringement, but there's no legal obligation to actually prove that in court before sending the request. Since the request is sent to the hosting service (e g. YouTube or GitHub) instead of the creator of the content, the content is often taken down before the creator even hears about the request. It is not in the hosting service's interests to fight such claims, nor look much more closely than to verify that the sender seems like they likely own the allegedly infringing content.
If you host your own content, yes, you're responsible for whether you take it down before being legally obligated to. Most people use a popular hosting service, especially for something like a video where any form of popularity will likely overwhelm their hosting. That's why people use services like YouTube, they don't want to deal with scaling if demand takes off.