What is called "Buddhism" is a whole range of different schools, sects, practices etc. that has in common that they are based on the teachings of Siddharta Gautama/the Buddha. But is the name "Buddhism" used as a name for these teachings anywhere else than in the "West"? How and when did the name "Buddhism" appear?
How and when did the name "Buddhism" appear?
I'm not an expert, but here's what I found this evening: the article SCHOPENHAUER AND BUDDHISM by Peter Abelson says,
When the tenets of Buddhism became known in Europe during the third and fourth decade of the nineteenth century
... and says that Schopenhauer referenced work by Isaak Jacob Schmidt.
Wikipedia's article for Isaac Jacob Schmidt says that one of his publications was titled,
Über die Verwandtschaft der gnostisch-theosophischen Lehren mit den Religions-Systemen des Orients, vorzüglich des Buddhaismus, Leipzig 1828
And this book Prisoners of Shangri-La: Tibetan Buddhism and the West quotes Isaac Schmidt as saying,
So according to Isaac Jacob Schmidt,
The peoples of that faith call themselves followers of the teachings of the Buddha.
According to Wikipedia he was living in Russia and studying Mongolian and Tibetan history.
His scientific work became noted after the publication of a work on the history of Mongols and Tibetans in 1824. ... Until his death in 1847, Schmidt published a multitude of works on Mongolian and Tibetan studies
I don't think I understand your question. It's simply an '-ism' made from what should be referred to as "Buddhadhamma". All schools agree that Buddha showed a way to nibbana which anybody can follow by critically understanding every step themselves. All schools also agree that their minute differences are mainly in the monastic rules and conduct which themselves are nothing but a raft used to reach a state, Buddha refers to Dhamma itself.
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Good question! The Buddha introduced his "religion" as (this) Dhamma- (this) Vinaya, Dhamma-Vinaya (Teaching & Discipline) and there is and was also the use of Buddha Sasana (the teaching of the Buddha).
The "ism"-isation is actually already against his teachings, especially such nuances as "my", "our" Buddhism, and indicates even much lack of respect, since the teaching of the Buddha, the Dhamma, is just for an Arahat his own and even so, a person who actually owns it, would not refer to it other than Buddha-Sasana.
As far as Atma knows, this name firstly came up about 1850 in UK (you find certain dates or orign and hints in older English dictionaries) and his actually a label of and everything that likes to use the Buddhas reputation for mainly wordily purposes, attachment, (wrong) views and identifications, so really not a good label if one seeks for the good teachings.
Buddhism is an English word that means the practice,system or doctrine of the Buddha, lit. "awakened, enlightened." Obviously the Buddha didn't speak English, but he did use a Pali phrase that means "followers of the Shakyan," referring to the Buddha's clan and territory (lit. "the able ones"), which means very nearly what is connoted by the English term. More usually, however, he simply refers to the Dharma-Vinaya, meaning "the doctrine and the practice," without specificially alluding to himself. Moreover, "Buddha" is really a collective term. The Buddha did not claim to present an original teaching specific to himself, but rather to an ahistorical or prehistorical lineage of perfected beings (buddhas) going back hundreds of thousands of years at least, which he equates with (but does not limit to apparently) the Vedic rishis. He states that this teaching is archaic, even primordial, and that he has only rediscovered it in the degenerate age, thus establishing a buddha dispensation (sasana) that will last 5,000 years (i.e., till 4600 CE approx.). The teaching of the ekayana, single or universal vehicle, is that all of these schools and sects are merely facets of one universal teaching that reconciles them all. Only the latter is actually Buddhism. There is no evidence whatsoever that the Buddha taught or thought in an exclusive or sectarian way.