Intraspecific will be the successful cross of 2 species of a genus. In this example, the 'x' shows up after the genus to show the cross at the species level. I may be using some tropical plant names that are unfamiliar to you, but maybe I can recall some more temperate NE perennial-type examples too:

intraspecific:
A natural cross between Bauhinia purpurea and B. variegata occurred in Hong Kong in the 19th century. The result is Bauhinia x blakeana. This follows the rule you mentioned--a new specific epithet. This new epithet demonstrates a new phenotype. It also makes it more practical than having to rewrite both spp. with the X in between!

There are also intrageneric crosses. I recall
x Heucharella (note where X is now)is one you may have heard of--Tiarella crossed with Heuchera. A more tropical example is X Ruttyruspolia which is a (seemingly natural) hybrid of Ruttya with Ruspolia. At more close inspection, it is a cross with Ruttya ovata X Ruspolia hypocrateriformis var. australis! Thus, whenever you write of this plant and it's cultivars, you need to correctly include the X as in X Ruttypospolia 'Phillis Van Heerden'.

I'm curious as to what you are disagreeing with. As I anticipate your question, perhaps this has to do with nurserymen screwing around with trademarked and cultivar naming for profit motives as compared to the International Rules of Nomenclature.

But I may have completely muddied the waters with my comments.

Good luck.

Ok, from what you are sharing, it may be useful (or confounding) to mention cultivars, var, and subspecies here.

Not knowing the true original phenotype of Syringa vulgaris (i'd assume it'd be a purple flower), it is doubtful that when the first say, white-flowering variation occured that anyone would have given it a natural-occuring forma epiphet or even a subspecies. Based on human preference for that white-flowered look, we'd just give it a cultivar name, right? No X anywhere in the epiphet.

So, if you are taking 2 horticulturally derived selected cultivars of a normally straight species (that has no ecotypes or natural subspecies), you in effect are creating a new cultivar is all...I believe even if you did cross a straight species with a natural forma or subsp. of that species, you are in effect creating a cultivar.

Based on that, in literature, you would list the parentage as say S. vulgaris 'Purple Glory' X 'White Snow' and thus yield S. vulgaris 'Gillmass' Surprise'. I don't believe it would be appropriate to add any X in the actual epiphet for this new cultivar as it occured simply as a playful cross (far removed from the original species, forma or subspecies if they had existed).

For personal reference, I think you could parenthetically post the parentage of the cultivars. Some hybridizers often will quickly give it a random 'U123' cultivar name showing it as an unnamed #123 cv. that may prove useful one day...it is back in their notes that the parentage of 'U123' would be listed. INteresetingly, because of the great genetic possibilities, you may again cross these same 2 parents and yield another hybrid that looks nothing like the first 'Gillmass' Surprise'. Then it'd be quickly labeled 'U124' and evaluated later.