Screenshots would be nice, but as far as I understood:
Basically, Compatibility mode is used to open documents created in earlier versions of Word.
You can read more about compatibility here
But, it is noticed that if you open a DOCX 2010 document created in Word 2010, but on another computer, it will be opened in [Read Only] as well as [Compatibility mode] or [Protected View]. I guess, this is made for protection from potential malicious includings. If you are sure about origin of the file you can do File - Info - Convert and then save the document.
Now, regarding Document Recovery
. It is not connected to Compatibility mode anyhow. Completely different feature. It was designed to save files when modified file will be closed without saving or Word itself crashes or to return to earlier version of the document.