Actually HSV-1 is certainly an STD when it results in a genital infection; it has exactly the same symptoms as HSV-2. For whatever reason, fewer people are developing immunity during childhood to HSV-1 as in the past, and as a result HSV-1 is causing more cases of genital herpes. Probably due to a combination of (i) the increase in oral sex combined and (ii) no pre-existing childhood immunity of the uninfected sexual partner. HSV-1 is now responsible for ~40% of all cases of genital herpes!! When I studied infectious diseases in the 90s, they were just starting to recognize how widespread HSV-1 mediated genital herpes was, but I did notice the older textbooks (70s/80s) on infectious diseases made the same distinction that you infer-- HSV-1 common virus that leads to cold sores, HSV-2 less common virus that leads to genital sores. It is no longer that simple. Also, HSV-2 is in fact also from the same general family of viruses that includes chicken pox/ shingles.
Viruses and bacterial infections can enter many places in the body. An STI has been traditionally been considered to be an infection that is primarily sexually transmitted. HSV 1 IS NOT, while I don’t disagree that it is sometimes transmitted during sex. But if being transmissible during sex is the criterion, then I suppose staph or any UTI, for example, would have to be an STI, too, because they could be transmitted during sex, as would also be the case with other infections that are not typically sexually transmitted.
The public health profession has become highly politicized, and claiming that HSV 1 is an STI is part of an effort to destigmatize the consequences of the promiscuity encouraged by the left. Frankly, saying, in effect, that a 6 year-old with occasional cold sores has an STI is despicable. In sum, lots of things can cause infections in the genitalia, but only a few of them involve what have been commonly understood to be STIs. Broadening the definition is political, unless something new has been discovered that is primarily transmitted sexually.