which is only meant for "backward compatible"
You've read too much into it. Using empty tags to represent empty elements is part of XML. It's not something that was added to XHTML for backwards compatibility with HTML.
the proper way to use a XHTML br tag is <br />
XML and thus XHTML allows it, so <br></br> is also correct, proper. You can make recommendations as to which syntax to use — and I'll agree with you that <br/> is less problematic — but you're out of line in saying one of them is incorrect.
a XHTML br tag is <br /> which has no "backward compatible" and is true XHTML
You have it backwards. <br /> is more "backwards compatible" to HTML than <br></br>. <br /> is interpreted as <br> by browsers, but <br></br> is interpreted as <br><br>. If your argument had merit, it would dictate that <br></br> is the proper way.