The answer is no, there is not a standard for this because the usage of the term 'Backdoor' to refer to 'accidental vulnerabilities' is incorrect. Therefore differentiating between intentional and unintentional backdoors is not something you can do, let alone standardise.
Anyone who says otherwise clearly has not been in the security game long enough to remember the ways in which the language and terminology in this space has evolved over time. Pick up any dusty book or 90's whitepaper on computer viruses and you'll see it plain as day... a backdoor is a method of malicious persistence.
I think the media are generally to blame for this one... us techies do our best to convey terminology to them in a way that can be consumed by the general public, and over time they seem to create their own media-jargon and/or warp the general public's idea of what a term means. For that reason I can 100% see how people can believe a vulnerability is/can-be a backdoor, likewise I can just about grasp how most people can't differentiate encryption from encoding.
In terms of a standardisation of the various sub-categories of backdoors, the closest 'global' method (sorry USA, but the whole World isn't listening to NSA/NIST/CERT guidelines) was created by the anti-virus vendors of the early 90's. Technically speaking, a backdoor is a type of virus.
I'd encourage you to go to https://www.virustotal.com/ and search for 'backdoor'. Check out any of the millions of uploaded virus samples and look at the different names that anti-virus vendors have given them. There are obvious naming conventions that appear and there is clearly some degree of naming standardisation going on, at-least within each vendor. Typical trends:
A backdoor thats only seen as a windows 32-bit binary? 'W32/Backdoor'.
How about a Backdoor, thats a windows 32-bit binary, that has its own cool name 'DoubleAgent'? Easy: 'Backdoor.Win32.DoubleAgent.c'.