To preface, the answer I will be writing here will primarily concentrate on the Qing perspective, as it is Qing history rather than Tibetan history I am familiar with. I for one don't think I could do the Tibetan perspective justice here.
A basic issue with answering this question is that it is necessary to define what 'China' actually means. I say this often but will say it again: 'China' is an English word that maps imperfectly onto a number of concepts in Chinese, and most contemporary historians of the Qing make sure to either define or provide strong context for how they choose to use the term in a Qing context. For the purposes of this answer I will be assessing three, kind of four, possible definitions:
'China' is synonymous with the Qing Empire.
'China' maps onto the Qing concept of dorgi, or 'inner' lands; alternatively, 'China' refers to a contiguous region with a majority 'Chinese' population, however defined. In other words, a region with a certain degree of political and/or demographic uniformity.
'China' refers to a region consciously identified as part of the national space of the 'Chinese', however defined.
Tibet didn't fall under all of these categories, and not all of the time, as I will stress; whether it even fell under a particular category at a given point is not unambiguous either.
When we say 'the Qing Dynasty' we need to do a bit of disambiguation and debunking. The Chinese term chao, which gets translated as 'dynasty' in the context of the imperial dynasty-states, is perhaps better translated as 'court', which gets across a much better sense of how Chinese statehood was conceptualised. There was not in fact one continuous Chinese Empire whose throne was passed from house to house. There were a succession of states that ruled – or aspired to rule – over a minimum set of territorial bounds that we can call 'China', but while they were understood as successors to each other, they were not inheritors of the same mantle. While it's fine to use 'China' to refer to any one state at a given point in time, it therefore becomes problematic to extend that across multiple states. If I were to lump the Han, the Tang, the Southern Song and the Qing all under 'China', I would be opening the doors to a lot of conflating on the part of my reader. This applies going forward in time as well: 'China' in the context of the Qing means something different than it does for the ROC or the PRC.
As, in this case, we are just referring to the Qing, we can safely use 'China' as a plain synonym for the Qing Empire. And in that sense, Tibet more or less can be said to have been firmly a part of the Qing Empire, though this was not a flick of the switch. I'd argue there are two main points at which Tibet can be said to have come under the empire: 1720 and 1793.
1720 was when the Qing expelled the Zunghars from Tibet and established a limited presence at Lhasa, consisting of a plenipotentiary official (amban) and a small garrison – small enough that its commander refused to commit it to the defence of Lhasa during the Nepalese invasion of 1789. The Ganden Podrang (the Dalai Lama-led government first set up by the Khoshuts in the 1640s) was still responsible for the ordinary governance of Tibet, and the Qing actually did much to cement Lhasa's authority over a number of regions in Tibet's environs, particularly Amdo and Kham. The Qing also, after the revolt of Chingünjav in 1757, started requiring that a number of lamas in Mongolia have their successors incarnated in Tibet to circumvent nepotistic lama appointments among Mongolian nobles. It is hard to imagine the Qing divesting territorial and religious authority to an entity that they would have considered outside the empire, hence why I would suggest that you can say the Qing considered Tibet to be within the scope of the empire from around 1720 onward.
However, during this time the Qing's power within Tibet was decidedly limited. The amban had no formal status within the Ganden Podrang, and the garrison, while still a significant protrusion of Qing power, was clearly not enough to actually defend Qing interests against a determined thrust from an outside threat. Things changed after 1793, although not necessarily hugely radically, as the Qing began to claim and exercise the right to veto lama selections – especially those of the Dalai and Panchen Lamas – and required that, in the event of any uncertainty, the selection process was to take place by lot through the Golden Urn, which entailed drawing the name of the new successor from a shortlist. While this did give the Qing a much firmer stake in Tibetan politics, it is worth stressing that this was a veto rather than a direct claim to control over appointments; the Qing were more interested in clamping down on nepotism and ensuring the credibility of lama candidates than they were in trying to directly appoint their own stooges, not least because there were at least some committed Buddhists – or, cynically, people with a strong motive to appear to appeal to Buddhists – among the occupants of the imperial throne. While there were things that didn't change much after the Golden Urn was introduced – indeed, the Qing only required that the Urn be used if there wasn't an obviously acceptable frontrunner candidate already – many Tibetan elites over the course of the nineteenth century increasingly came to see the Qing as an oppressive overlord as opposed to a cooperative patron.
So in that sense, Tibet was definitely part of the Qing Empire, if not after 1720 then after 1793, and, if as above we have we defined 'China' as being 'the scope of whichever China-ruling state was in the ascendant at a given point', then Tibet was 'part of China during the Qing Dynasty'.
In Manchu texts, the Qing often considered their empire to consist of two components: dorgi ('inner') and tulergi ('outer'). It is worth stressing that this was not a statement of relative importance, but one of geography and also of administrative structure. The dorgi lands consisted of China and the Northeast (or 'Manchuria'), regions that had maritime connections, were dominated by sedentary agriculturalists, and were administered through regularised, hierarchical bureaucratic systems (although China and Manchuria were governed through separate bureaucracies). The tulergi lands consisted of Mongolia, Tibet, and Qing Turkestan ('Xinjiang'), which were landlocked, generally had a larger proportion of nomadic pastorialists and were historically politically dominated by them, and which the Qing governed through the Lifan Yuan or Bureau for Administering the Outer Regions, which was generally staffed almost exclusively by Manchus and Banner Mongols. For the most part, the Lifan Yuan's officials simply supervised local government and dealt with disputes between members of different Qing subject populations, with actual regular governance being carried out through various local elites, but retained a considerable degree of extra-ordinary authority. In Mongolia, these local rulers were the jasak chieftains; in Turkestan, the hakim begs; and in Tibet, the Ganden Podrang.
The conception of the exact relationships between the various bits of the empire varied from emperor to emperor, but it is fair to say that dorgi and tulergi were generally seen as relatively coherent zones for the most part. There was certainly a reasonably consistent belief that the Han and Manchus, while distinct peoples, belonged to a common paradigm of sedentary peoples that the Mongols and Tibetans did not, and until the mid-nineteenth century the distinct administrative structures for dorgi and tulergi remained largely unchanged. In addition, movement of peoples was rather restricted going from one to the other. Han migration outside of China proper – including into Manchuria – was heavily clamped down on, and Han officials would be sent to Xinjiang as punitive assignments for misconduct or incompetence, rather than as a conventional career posting. But Tibetans, Mongolians, and Turkic people moved comparatively fluidly through the dorgi zone: Tibetan lamas served in Mongolian monasteries, certain Mongol tribes, particularly the Khoshuts, lived in Tibetan-ruled Amdo, and the depopulated region of Zungharia, devastated by the Zunghar Genocide, soon became a melting pot of Mongols, Tibetans, Turks, and Tungusic peoples like the Manchus and Xibe.
However, the bounds of the Han Chinese bureaucracy would expand considerably, particularly after 1850. By the eve of the First Sino-Japanese War in 1894, Han officials had supplanted indigenous chiefs in Yunnan, Guizhou, and Taiwan; the Manchu bureaucracy in Manchuria had largely become sidelined by the officials responsible for Han affairs, thanks to massive movements of Han colonists into the region from the 1850s onward; Xinjiang was taken out of military rule and became a province of the empire in 1884. Latterly, Inner Mongolia was subjected to Han colonialism during the New Policies period after 1901. Tibet, though, largely avoided Han colonialism during the last decades of the Qing, and was never placed under the authority of the Chinese bureaucracy proper.
So, Tibet did not fall under either the administrative umbrella that encompassed 'China', nor was it subsumed by the colonial ambitions of the Han, during its time under Qing rule.