Keep in mind that inoculation and the smallpox vaccine are not the same thing. In fact, smallpox inoculation has been known and documented in China at least since the 16th century, and possibly earlier. Inoculation was done by exposing the patient to small pox postules, making it considerably more dangerous than Jenner's vaccine, which used the comparatively harmless cowpox. There was a significant chance that the patient would actually develop a smallpox case. Although it was still on average considerably safer than remaining unprotected against smallpox, that risk made it difficult to convince families to inoculate their children, keeping rates of adoption low.

Reports about the Chinese practice of inoculation reached the Royal Society in London by 1700. Similar practices appear to have existed elsewhere in Asia, but are not as well documented - the Royal Society received other reports of inoculation in the Caucasus in 1714 and in India in 1767. However, the Chinese version is the only one with surviving local-language medical texts, given us both the earliest and most detailed descriptions of inoculation. Even then, none of the surviving Chinese texts claim to have invented inoculation - the earliest text is a discussion of inoculation side-effects, and slightly later detailed guides on how to conduct inoculation start to appear. This makes it hard to be sure exactly when inoculation was invented - or even if it originated in China at all.

By the time George Washington inoculated his troops, smallpox inoculation had been widely known in educated circles in Europe for some time, if not always widely practiced. Starting with the major smallpox epidemic in 1721, elites in Western Europe (notably Voltaire) began campaigning for inoculation, however for the reasons mentioned above, they were not always successful in convincing local doctors to practice it.

Jenner's vaccine is a different matter. Jenner discovered that you could essentially perform inoculation with cowpox lesions instead of actual smallpox postules and achieve the same results at much lower risk. While the technique still poses the risk of illness, the patient would develop a case of the nearly-harmless cowpox instead of potentially deadly smallpox. As a result, smallpox vaccination didn't really become widespread until Jenner's cowpox based vaccine, and it is Jenner's vaccine that is usually considered the pre-cursor of modern vaccines rather than the earlier practice of inoculation. It probably also helps that Jenner's invention can be assigned a (European) name and a specific date, which usually makes inventions more memorable in the public eye compared to practices which originated in Asia at an unclear time. As an aside, cowpox only exists natively in Europe, which is probably one reason why none of the Asian societies practicing smallpox inoculation ever realized that cowpox could also grant immunity to smallpox.

Sources:

Needham, Joseph. (2000). Science and Civilization in China: Volume 6, Biology and Biological Technology, Part 6, Medicine. Cambridge: Cambridge University Press. p. 134.

Voltaire (1742). "Letter XI". Letters on the English.

Holwell, John Zephaniah (1767). An Account of the Manner of Inoculating for the Small Pox in the East Indies with ... Observations on the ... Mode of Treating that Disease in those Parts. London: T. Becket & P. A. de Hondt.