There already is a "universal" term, it's called having an "NRTL Label"; NRTL stands for Nationally Recognized Testing Laboratory. That is how OSHA words it and technically is the ONLY type of label that is required by any codes, most of which point to OSHA as the defining
list of acceptable NRTLs. So nowhere does anything say "UL" other than that list, and UL is just one of the NRTLs listed. Some larger jurisdictions such as Los Angeles and New York have their own lists of which of the OSHA NRTLs are acceptable, and may even include some that are not on that list. But nobody says UL has an exclusive. UL is just the most well known of the NRTLs, and in fact most of the others don;'t even bother to have their own testing standards, they just buy a copy from UL and use them.
UL is a company with a product to sell, namely their testing services and the development of testing standards. They are not a government agency, nor in any way affiliated with a government agency. Their name comes from "Underwriter's Laboratories", although more recently they changed it to just "UL". That was Underwriters, as in INSURANCE underwriters. UL originally was a testing lab serving the nations largest fire insurance carriers as a way of them having some way of knowing if equipment that their customers were installing was going to cause (or prevent) a fire or not. It was put together after the Chicago Columbian Exposition, where tesla and Westinghouse first displayed their AC electric lights for the world. Chicago, still remembering the Great Chicago Fire, had hired the guy that later founded UL to inspect and test the way those lights were connected and make sure they were not going to start the Chicago Fire #2. Afterward, he started a lab to do testing like that for others and insurance underwriters jumped on the idea.
So in your original post, you mentioned "We are going thru UL testing with someone other than UL labs." That would be impossible. You are going through NRTL listing, but you will not have UL listing, nor will it be UL doing the testing, nor will UL be in any way interested in discussing this with you, because you are using a competitor of theirs. That's like saying "I am using a GE breaker, except it is not made by GE, but I want GE to stand behind it". Ain't gonna happen...
That's not to say the other NRTLs are not good, but when your customer says they don't want to accept it, they have that right. You will on occasion find that an insurance company, in underwriting a fire insurance policy, will have a rider saying something to the effect of, damages in a fire determined to be caused by a device that could have been UL listed but was not, may not be covered. The insurance companies, being private enterprises, have every right to chose who's standards and testing they will or will not accept.
An ELECTRICAL INSPECTOR however cannot say that. If their jurisdiction has a requirement for an NRTL, and they defer to the OSHA list or their own list, then anyone on that list will suffice.