So why does Virginia law specifically tell inspectors to "fail if"...
"1. Lamps are not of an approved type (DOT or SAE-P2, P3, PC)
P2, P3, and PC are nothing to do with fog lamps; those are marker and clearance lamps. But I see what you're asking. "Approval" in that line refers to lamps the State of Virginia accepts, which has been irrelevant since 1968. Prior to that time, each and every vehicle lighting device -- every brake light, every turn signal -- had to be approved by each and every state. It was a big, expensive hassle that ended when Federal Motor Vehicle Safety Standard 108 took effect, because the law that set up the Federal Motor Vehicle Safety Standards holds that States are not permitted to enforce their own standards more stringent than the Federal standard that applies to a vehicle or item of vehicle equipment. That means states are not allowed to pick and choose which Federally-acceptable items of vehicle equipment they will or won't allow on vehicles registered in that state. That's a very, very good thing, because many states have vehicle lighting codes that are a mess of poorly-written, extremely obsolete text; Virginia's is one of the worst (Dan Stern passed around a running-commentary analysis of Virginia's vehicle lighting code a number of years ago. I thought I'd kept it, but I'm not finding it; if you're interested, write and ask him.) If Virginia's vehicle equipment code hadn't been rendered null and void by the Federal standard, exactly zero percent of the cars presently on the road would be legal, because they all have headlight bulbs producing more than 32 candlepower (and a whole bunch of other stuff not allowed by VA's antiquated code).
If DOT and SAE don't codify lights then who does?
The issue at hand is "approval". What I was pointing out to the OP is that "DOT approved" and "SAE approved" don't mean anything on any lamp, and "DOT" on a fog lamp doesn't mean anything.
SAE is not a regulatory body and has no means or authority to approve anything. SAE just issues technical standards, supposedly representing the best practices in whatever the standard is about (fog lamps, turn signals, headlamps, etc). These technical standards do not have any force of law, except to whatever degree a political entity adopts them. The Federal Motor Vehicle Safety Standards incorporate some (but not all) provisions of some (but not all) SAE technical standards for vehicle lights, and while SAE standards are updated periodically, every four to six years, these updates are not automatically acknowledged by NHTSA, the agency within the federal Department of Transportation responsible for the Federal safety standards. If you read the Federal lighting standard, you'll find it contains numerous references to very old versions of SAE standards, going far back into the '60s. That's a shame, but those are the provisions that have the force of law.
There is no such thing as "DOT approval". NHTSA and DOT are not legally empowered or authorized to approve vehicle equipment; that's not how the American system works. Instead, the responsibility lies completely with the maker or importer of the equipment. The "DOT" marking on a headlamp does not mean DOT or anyone else has approved it, it means the manufacturer or importer has decided the headlamp meets all the legal requirements. "DOT" on the lens is the maker/importer's own assertion, or certification, that the headlamp complies with the law. There is no particular pre-test required; lamps do not have to be sent in to the DOT or anyone else before the marking can be applied -- the maker does whatever they feel is necessary to satisfy themself that the lamp meets the standard. Reputable companies put their lamps through rigorous tests. Shady companies laugh and put "DOT" on the lens, gambling (often a winning bet) that their lamps will never be checked out formally. If an investigation is carried out and a headlamp marked "DOT" is found to be noncompliant, NHTSA can force the maker/importer to issue a recall and can issue penalties in the form of fines. Very steep fines, assessed on a per-violation basis, with each sold headlamp counting as one violation. But these enforcement actions are rare, and even when they do happen, the target is often a slimeball "company" consisting of not much more than a warehouse for unloading and stocking containerloads of pretend-headlamps from China, and a marketing department to promote them. Get smacked with penalties? Declare bankruptcy, disappear for a week, open up shop under a new name, continue business as usual.
You'll notice I've been talking about headlamps for the last paragraph or two. That's because headlamps are the only lamps required by law to have a "DOT" marking, and therefore the only lamps on which "DOT" actually means anything. There's nothing stopping anyone from marking other lamps "DOT", but it means whatever the maker decides it means, which could be anything from "It means this lamp meets the applicable provisions of Federal Motor Vehicle Safety Standard 108 as they appeared in the official legal text as of the date the design was locked in for production" all the way to "It stands for Dogs Over Tampa".
You'll frequently see "SAE" followed by letter/number codes, followed by a 2-digit year code, on many vehicle lamps. These markings aren't legally required, but they're allowed, and they mean that the maker says the lamp meets the requirements put forth for [whatever function codes are present] as of [whatever year is listed]. For example, you might find "SAE AI6RSTP2 97" on a multifunction taillamp. There's a breakout chart of the SAE codes on
this page which explains what all those codes would mean. But it still doesn't mean or imply anything about anyone's approval.
Fog lamps are not regulated Federally, so states are free to regulate them however they wish, from no regulation at all clear on up to no fog lamps at all (not allowed to be used in that state). Front fog lamps can be marked "SAE F [year]" or "SAE F3 [year]" -- there is no legal requirement for any such marking, though some states say you can only have fog lamps with an SAE marking.
As you can see, it's more complicated than it seems on its face, and the lighting aftermarket's many scam artists are safely betting that most people will think "DOT Approved" or "SAE Approved" or "SAE/DOT Approved" is some kind of a meaningful assurance.