Anyone who's French, particularly the old French nobility of the sword.
OH COME ON.
What are you, "old noble" historian from XVIII? Aren't you supposed to be all guillotined?
For the Franks, there's two definitions at my sense.
Ethnic : The Frankish apport is clearly more important in the zone between Loire/Rhine/Saône. Not only you have more toponimy elements (as the most famous L'Ile de France, from Liddle Franke "Little Francia") but in the names of nobility you have far more typical and localized Frankish names until a quite recent date (by recent, i mean the XII). The gallo-roman elements was not really present in these zones (contrary to the west, and obviously the south) and they were quickly assimilated into Frankish nobility (as the quick constitution of the great domains of Austrasia seems to show).
Finally, the greatest Frankish families, the royal then imperial fisc, the palaces are almost all situated in this zone.
So, maybe you had another Frankish settlement in border zones (Frisia, Alemania), but that not overnumbered the local nobility. But, you can consider the Austrasia, plus a third of Neustria as being ethnically populated by Franks, and as the original population wasn't that big (Gaul at the VI, it's maybe 8 millions of inhabitants) Frankish identity overruled.
That's is the second definition : identification.
Since the VI, chronicles made a clear distinction between the Franks (population of the north) and Romans (maybe Gascons) in the south. These texts are comically modern regarding xenophobia and stereotypes. Little by little, the northern population began to assimilate itself to the Franks, in their law basis, customs almost all except 1. Religion 2.Language that will be eventually assimilated by Franks.
At the contrary the southern population showed a more loyalty towards the local dynasties, even if they were Franks as the Guilhemids.
It's not for saying that the southern romans and the gascon always fought the Franks, no Provencal patrices, dukes of Gascony as Lupus II, some monasteries allied themselves with the Pippinids : but they were always a clear distinction regarding identity.
A Frank, or a Frankish noble of VIII (or even from XIII after the crusade) would say he's a "Frank mann", which gave the occitan word "Franciman" for name french or "devoyed" southerner.
During all the Middle-Ages, and beyond, you'll have 4 great regions :
West (Neustria until Paris, Normandy, Maine, Berry) where people can call themselves Franks but where the gallo-roman element is really present (more alleus in Maine by exemple)
Francia (all between paris, Rhine, Loire and Saône) : with a far lesser important gallo-roman element, where Franks mix with a more close (for them) Celtic element.
Aquitaine-Provence : A gallo-roman element that is extremly present, and even if it knows an eclipse during the X-XII, it preserves carachteristic traits, and "returns" to roman traditions regarding law since the XII thanks to the proximity of Italy.
Others : Britanny, Gascony, Navarre : all that is not entering in other zones.