There seems to be some confusion about the source of the name "China" in Western usage. The question has to be divided into two parts:

2. How did it become adopted in Western usage as the standard name for the country, since the Chinese themselves have used many other names to identify themselves, their country, their language, etc?

The first question is easily answered. The name ultimately derives from the Qin (pronounced Chin). Qin was one of the early "states" of pre-Imperial China, and was the first to conquer all the others and unify China into a single state in the 3rd c BCE, which was then superceded by the Han Dynasty.
https://en.wikipedia.org/wiki/Qin_dynasty

Later the name was taken by a later dynasty: On the Later Qin (384-417 CE), https://en.wikipedia.org/wiki/Later_Qin

During the Later Qin, many Buddhist texts were translated. For instance, the first text in the Taisho edition of the Buddhist canon is the Dīrghāgama, tr. in 413 CE, which was during the Later Qin Dynasty 後秦 (houqin). In its preface, one finds terms like:

秦 Qin (referring to China),
大秦 Daqin = Great Qin (referring to China, and the source of the Sanskrit Mahācīna)
秦國 Qinguo = Country of Qin.

The term Qin occurs nearly 12,000 times in the Chinese Buddhist canon (though sometimes referring to something other than China or Chinese, e.g., 拘樓秦 Julouqin = Krakucchanda, one of the Seven Buddhas of the past.). The self-reference by Chinese to their own country and culture and rulers, etc. as Qin, Daqin, etc. continued through many other periods, not just these dynasties. Xuanzang (Hsuan-tsang), e.g., in the 7th century, refers to China as Daqin.

The second question is murkier. Secondary sources complicate the problem. Some seem unaware that for many centuries the Chinese referred to themselves as Qin and Daqin (among many other designations), since that is not a modern name in present usage, and writers familiar with current usage seem unaware of the name's history. Some cite Persian as the route by which the term entered western usage, sometimes attributing its introduction to the West through Marco Polo. Some additionally speculate that the term came into Persian from the Sanskrit.

It is worth noting, however, that the Hebrew term for China -- already in the Bible -- is Sin (סין), and the same goes for Arabic (al-Sin, الصين). The Persian Chin (چین) could just as easily have come from Semitic sources as from Indian sources.