There is a term used in Japan for English “borrowings” that in fact originate in Japan, which is 和製英語, wasei-eigo or “made-in-Japan English”. This term is typically used for words like ペーパードライバー peepaa doraibaa, meant to be “paper driver” or someone who has a license but doesn’t drive (why?), or アイドリングストップ aidoringu sutoppu, which any junior high school graduate should know couldn’t possibly be interpreted as “stop idling (your car)”. As should be obvious, I regard the term wasei-eigo as stupid, but revealing of some interesting assumptions about what it means for a word to be part of a particular language.
If someone says biscotti is Italian, not English, s/he is heuristically substituting a question of origin for one of current membership in the Italian language. Of course by this definition most of the functional vocabulary of any language is not part of that language – words like Tuesday, vulgar, and robot all have non-English roots in the recent historical record, but who would be willing to say that these words are not English? At least for this reason, foreign etymology cannot be our test for whether a word is English or not.
The term “English” is an abstraction of the linguistic knowledge of a somewhat arbitrarily delineated group of people. It includes all the words that group uses, even ones that it uses while considering them foreign or unconventional, like biscotti, sushi, or blerg. A monolingual Japanese speaker making up a word and calling it English makes as much sense as Mitch Hedberg’s buying a 2-bedroom house and calling it a 4-bedroom, 2 of which are in another building. Hence wasei-eigo are part of the Japanese language – in fact it would be hard to call someone a competent speaker of modern Japanese if they didn’t know many of these words. The term wasei-eigo should be replaced with something like 英語かぶれの日本語 (Wannabe-English Japanese) or 擬似英語 (false English) – I personally will start correcting people and having very awkward conversations on the topic one day soon.