I will start by giving a quick plug to /r/ChineseHistory because there are some educators over there who might have a useful perspective on this. The answer you will probably get is that the Qin and Han are "classical", the Shang and western Zhou are "ancient" and considerable disagreement about where to put the Spring and Autumn and Warring States periods.

You are correct that the distinction seems subjective, because ultimately it is, in two ways: for one, the terminology is an importation of terminology from the west in an attempt by western and western trained scholars to find equivalence in Chinese history. This does not mean it is useless and false, terms like "medieval" and "early modern China" are still used today because they usefully convey certain ideas, but that does mean they fit somewhat uneasily. I have a book called Medieval Chinese Warfare which implicitly makes a very useful case for the aptness of the word, but that does not mean you will find an equivalence in the court of Charlemagne or Eleanor of Aquitaine for the position that Tang era Chang'an holds in the Chinese popular consciousness.

The second reason is that even in the west the term--and its distinction with ancient--is vague and fraught. Classicus in Latin means classy or high status, engaging in "classical learning" means engaging in the education of the upper class. Which means, for complicated reasons, the literary study of Greek and Latin texts from the period before about 500 CE. This term then becomes conflated with the cultures and civilizations that produced it, such that we might talk about "classical civilization" much as we talk about "Vedic civilization" in India or, indeed, the "Spring and Autumn Period" of China. But even classical studies has its classical studies, such that "classical Greek literature" is that produced in Athens from about 500-300 BCE, and "classical Latin literature" is that produced from about 80 BCE-20 CE. These are based on events (the Persian Wars to Alexander and the Late Republic and Augustus respectively) but if it all seems a bit arbitrary, well, it is. The point is that "classic" is entirely a retrojection, it is fundamentally the period that people look back on as "the acme".

So you combine these two vagaries and you can understand how “classical China” is itself rather vague, it is best thought of as referring to the period of China that is thought to occupy a similar position in “the story of China” as Greece and Rome do in “the west”. While the comparisons between the Roman and Han imperial periods are fairly obvious and in fact constitute something of fruitful subdiscipline today, beyond that it gets a bit strained. For example, it is possible to compare Plato and Confucius in terms of position within their respective philosophical histories, but they did not occupy similar social or political worlds at all. And while the Hellenistic period and Warring States period have certain similarities in that they were the crucibles of later state development, the comparison is hardly 1:1.

And of course over all this there is a certain question of whether it is better to simply stick with the traditional Dynastic periodization. For example, much of what I wrote here comes from jacques Gernet’s A History of Chinese Civilization, an influential textbook from the 70s that forcefully made the case that as artificial as they are, terms like “classical” “Medieval” “Renaissance” etc allowed us to get away from the strictures of purely dynastic categorization, but if you actually read the book he never really gets away from it.

Qin (221 BC–206 BC) is the start of Imperial China, whereas Qing (1636–1912) marked the end of Imperial China.

Within the umbrella of Imperial China, Qin and Han dynasties (221 BC - 220 AD) are commonly considered Early Imperial China marked by the emergence of a highly centralized, bureaucratic governance framework across a vast multi-ethnic territory, known in Chinese as 大一統王朝 (“grand unified imperial dynasty”). Many historians (as well as most Chinese history textbooks) also group the subsequent three and half centuries of (mostly) continuous state of warring factions, competing claims of imperial authority, and cultural convergences which spans across Three Kingdoms, [Jin](https://en.wikipedia.org/wiki/Jin_dynasty_(266%E2%80%93420/) and Northern and Southern dynasties also as part of the Early Imperial China, whereas others would categorize this period of disunity (220 AD - 589 AD) as a separate transitional period.

The re-unification of China under the Sui dynasty 589 AD also marked the beginning of the textbook category of Middle Imperial China. Just like Early Imperial China, modern day Chinese historical taxonomists often sub-divide this period into a "grand unified imperial dynasty" era between Sui (589 - 618), Tang (618 - 907) and Song (Northern, 960 - 1127) dynasties, and a later era of division and changing ethnic relations from the Jingkang Incident to the end of Yuan Dynasty.

The Late Imperial China is a less messy and contested category, typically consists of the last two “grand unified imperial dynasties" of Ming (1368–1644) and Qing (1636–1912). However, history textbooks in China typically group everything between the beginning of the First Opium War in 1839 and the end of the last Chinese Civil War in 1949 under the category of 近現代史 (“near modern China”).

The term "classical China" is NOT a commonly used general historical label/category within China. It is however used to denote a particular moment of Chinese philosophical and literary history, of which the classical period of Chinese intellectual roughly corresponds to 春秋戰國時代 which encompasses the Spring and Autumn period and the Warring States period, between 776 BC and 221 BC.

Pre-Qin China or 先秦時代 is the most commonly used umbrella term covering the period prior to Qin's unification in 221 BC all the way up to the Shang (c. 1600 BC – 1046 BC). The upper limit of Pre-Qin China is defined by the earliest confirmed written Chinese historical record, which is so far the Oracle bone script from Shang period.

Whether Xia belongs to Pre-Qin China is a still debated issue in China. There are numerous archaeological urban cultures discovered throughout China that roughly correspond to the quasi-legendary Xia period, but Oracle bone texts made no mention of Xia, and contemporaneous Xia written records have yet to be found. For this reason, more often than not, Chinese history textbooks places Xia dynasty along with other early Bronze Age cultures into the category of 史前時代 "Pre Written History China."

I hope this helps.