So first, lets cover the name of China. Or rather the more important question of what's "Chinese."

The difficulty of this, as evidenced by quite a lot of historians is that the "China" we call today, really doesn't exist until the Qing Dynasty. For example, the Qin and the Han Dynasties don't actually occupy quite a large portion of what the geographical borders of China are. While the Tang at a certain point stretched all the way to the Tarim Basin.

Not to mention that China itself has had quite a difficult time in determining who's "Chinese," for probably the entirety of its existence. Solutions like the separating the people as "Han and 55 Ethnic Groups" have at best been..mildly successful. At any rate, prior to the Tang-Song usage of Zhongguo or "中國" was not commonly found in texts contemporary to the time until near the end of Tang.

Instead we see "hua" or 华 used much more commonly. In other words, the notion of "middle kingdom" or "zhongguo" really doesn't exist until much later after the Qin Dynasty. As for "Han" people. This I'll probably leave to more knowledgeable members. However I will posit that the reference to "Han-Ren" or "漢人" is rather complicated considering that the idea of a "Han Person" most likely emerges as a reaction to the existence of the negative. That is to say, "漢人" probably doesn't emerge until after the Han Dynasty as a qualifier for identity. Though it very well may have existed if only because of the expansionist policies of the Han against the Steppe people.

But back on the notions of China. The usage of "Hua" or 华 are still prominent, though most prefer to use "Huaren" or 华人 or "Zhongguoren" "中國人" today. What's interesting though is that the Cantonese term for "Chinese" is "Tong-yan" or "唐人," a direct reference to the Tang Dynasty.

The word "China" in English on the other hand, is an even weirder term. As "China" is a foreign designation from the little I've gleaned from /u/bitparity. So as far as its origins it most likely doesn't have any basis in the Qin or Han Dynasty.

The differentiation between the people and the land however I would argue, most likely begins at the end of the Tang Dynasty and solidifies itself during the Song. Considering the rather dramatic loss of territory at the end of the Tang and the Northern Song Dynasty, poetry and literature (including Maps from the Song Dynasty) indicate that there was at the very least a sentiment of irredentism. A feeling of wanting formerly lost territory (in this case by the invading Khitans and Liao People) that formerly "belonged" to them. As a result we see the term "zhong-guo" being used in much of the literature pertaining to the territory of the Song, especially in relation to the "outer" or "lesser" territories. From at least the literature there definitely was a stigma and "Song-centric" view that existed, perhaps due to a loss of territory, but perhaps due to specific classifications that were made during the Tang.

Which brings me to one potential reason as to the promulgation of the term "漢人". Though I am unable to explain the context, I am however able to provide an explanation as to its widespread use during and after the Tang. Marc Samuel Abramson in Deep Eyes and High Noses: Constructing Ethnicity in Tang China suggests that despite the cosmopolitan nature of Tang China, there were still many different classifications of foreigners. In the case of the Tang, this included Uighurs, Turks, Sogdians, Liao, Xiongnu, Iranians among many other cultures. Incidentally all of the aforementioned were described as distinctly different, generally uncultured and disparagingly compared to beasts. Though perhaps not entirely new as the Han described the Xiongnu as "horse people" and "wild and unkempt" with "bad manners" whereas the Tang applied those descriptions to other Steppe tribes, among a slew of rather unfavorable descriptions there is however a rather interesting component to this. According to Abramson, and Edward H. Schafer's Iranian Merchants in Tang Dynasty Tales the comparison to non-human objects also carried an implication that culture, as well as definition necessarily derived from upbringing.

That is to say that the Tang considered "bad manners" and culture to be a integral part of classifying what was and was not "Han." Perhaps more pertinent to the point was that the nativism of claiming "Han" by the Tang Dynasty was most likely impossible to prove. After all a span of 4 centuries would have very much eroded lineages or at least obscured them. On the other hand, these classifications of "Han" by the Tang also implied that "Chinese" was not necessarily defined by appearance. In a few descriptions, there were people who looked "Han" but acted unlike what the Tang conception of "Han" was, implying that appearance was accompanied by culture. Likewise, Tang writings considered foreigners who acted with Tang mannerisms and culture decidedly "Han." Though they may still disparage on appearance, culture seems to have played a part of it.

This classification becomes much more important by the Song Dynasty. Considering the rather insular nature of Song politics and worldview we then see the aforementioned development of the term "Zhongguo" or "Middle Kingdom." As mentioned before, the insular feelings of the Song very much places both "Zhongguo" and "Han-ren" as important identifiers especially with regards to foreign policy as their of dealing with the Khitan and Liao tended to be colored by resent as the Song paid tribute to keep invaders from invading their territory. What we see from the tribute system and the political rhetoric advanced echoes what the Han had used to justify annual tribute to the Xiongnu during the reign of Emperor Wu, which was that the "Five Baits" would "pacify" the "barbarians." The Song applying that similar rhetoric towards the Northern Conquerors added a component of those tributes perhaps "sinicizing" the Northerners. Indeed considering that much of the lost "Song" lands were still administered through Chinese bureaucracy, there may have been a rationale behind the rhetoric. However because of the retention of Song bureaucracy within those conquered lands, the idea of "Han" culture retaining itself would lead to an affirmation that there was an innate from birth difference and cultural superiority with those who fell under the "Han" category.

Sources:

1. Edward H. Schafer, Iranian Merchants in Tang Dynasty Tales

2. Marc Samuel Abramson, Deep Eyes and High Noses: Constructing Ethnicity in Tang China

3. Michael Nylan The Rhetoric of "Empire" in the Classical Era in China

4. Paul S. Ropp China in World History

5. Assorted Song Dynasty Era Maps that were photographed.

The etymology of China is far from settled. According to the Oxford English Dictionary:

Not a Chinese name, but found in Sanskrit as Chīna about the Christian era, and in various modified forms employed by other Asiatic peoples. In Marco Polo Chin, in Barbosa (1516) and Garcia de Orta (1563) China. So in English in Eden 1555. (The origin of the name is still a matter of debate. See Babylonian & Or. Recd. I. Nos. 3 and 11.)

There is reason to be skeptical about the oft-repeated derivation of China from Qín 秦, the first unified imperial dynasty of China. According to Nicholas Ostler in Empires of the Word (2005), the Sanskrit Cīna चीन "applied mainly to the area of Tibet, though also on occasions included Assam and Burma (Sircar 1971: 104–5). China as a whole was known to the Indians as Mahācīna, 'Great China'".

Sircar, D. C. (1971), Studies in the Geography of Ancient and Medieval India, Delhi: Motilal Banarsidass.

I have also heard that the name "Zhongguo" didn't actually originate as meaning "middle kingdom," the derivation that's commonly (traditionally?) attributed to it. Is that true? Can anyone speak to the source and diffusion of the name, as well as those of the current popular understanding of it as "middle kingdom"? This may be a better question for another subreddit, but I like this one for its rigorous standards for answers.