I'm at work right now, but I'll do my best to answer your question after work. Do you happen to remember which source said that "knight" didn't come to "nobles serving as elite mounted cavalry" until the Hundred Years War? That would help to answer your question.

I'll just give a reference from a single non-European source that you might be interested in.

In Usamah ibn Munqidh's 'Book of Examples' (or something of that sort), an anthology of stories, mostly taken from his own life, he uses the word fāris (horseman) to refer to both muslim and Frankish horsemen (translator's note can be found on page 42). This is kind of interesting as he has a paragraph describing the social position of the Frankish 'knights' (page 93), which he does not expect his audience to be familiar with, and he uses transliterations of Frankish words for some concepts ('sergeant' for a footman, and 'madame').

I'm going to tackle this in a slightly different way than /u/Hergrim, since they're coming at the question from only one side of a debate that's long, complex and ongoing. They are, however, totally right on the main question: there were absolutely terms for mounted (socially-)elite warriors before the 14th century. They were for the most part existing Latin words (for Latin is the language of all our sources on this in the early period) which acquired extra connotations and began to be applied to aristocratic mounted warriors from around 950. Miles (which originally just meant 'soldier') eventually became the standard Latin for 'knight', but equite (horseman) and derivatives, and caballarius (again, meaning 'horseman' - especially common in Southern France) were equally important for describing this emerging class.

The question of whether men described as milites (or similar) in the 10th and 11th centuries can really be called 'knights' however, is heavily contested by historians. As /u/Hergrim pointed out, many historians don't see knighthood as a noble institution until the middle of the twelfth century. The argument on this side is basically that the 'knights' who show up in the late 900s are one rung below the nobility - desperate social climbers getting ahead by acting as muscle for the nobility by oppressing their peasant peers. By the 1100s, they were increasingly accepted by the nobility as part of the aristocratic stratum, as the nobility both adopted some of their martial values while simultaneously 'civilizing' them by the application of chivalry and chivlaric codes and rituals to knighthood. This view is fundamentally based on the work of Georges Duby and his student Jean Flori; it plays a vital role in Poly and Bournazel's infamous "feudal mutation".

This view of the development of knighthood has come in for serious criticism since the late 1990s however, particularly from Duby's enfant terrible Dominique Barthélémy. He argues that rather than up-jumped hyper-violent peasants, milites were elites from the moment they appeared; even if the title wasn't taken up by the very top rungs of the aristocracy, the ethos was shared among the entire noble class. Barthélémy, alongside others such as Jinty Nelson, has even argued that the emergence of knighthood can be traced as far back as the 9th century, when liturgies for blessing arms and ceremonies of arms presentation were the precusors of 12th century dubbing ceremonies. (Flori, and those writing in his shadow, recognise these texts, but see them as part of a "prehistory of knighthood", but not the full thing.)

The problem with sifting through these varying interpretations of early knights is that each vision is tied to a larger version of the 'story' of the central medieval period to which the author subscribes (usually one side or the other of the 'feudal mutation' debate, or in the Bachrachs' case their left-field views on early medieval/super-late Roman warfare). I'm inclined to agree with Barthélémy that knights were noble from the beginning: people decided to call themselves knights not so they could ride a horse to a fight, but to claim/maintain a certain social status. The full flourishing of long-gestating ideas of the tripartite division of medieval society (into fighters, workers and clergy) at just about the same time as the emergence of the title miles seems to me confirmation that it was a status symbol.

So, there was definitely a word for 'aristocratic mounted warrior' by the year 1000 at the latest. Whether that means he was a 'knight' (without the full chivalric trappings, dubbing, etc.) is a matter of definitions and is very much still up for debate.

If you're interested in reading more on the subject, Richard Kaeuper's Chivalry is the best and most up-to-date introduction on the debate (esp. chaps. 3-4). It's also part of the excellent Cambridge Medieval Textbooks series, which means it's actually affordable for people who aren't institutional libraries!

As a follow-up question, how did the basic concept of the knight differ - militarily, culturally, or socially - from one medieval nation to another?