Can someone tell me when people of Eastern Asian ancestry got a monopoly over the word Asian?
Back in the day, people would call others from China, Japan etc "Oriental" but now that word has fallen out of favour, some even deem it offensive. Oriental comes from the Latin word meaning East. So in English speaking countries, Oriental is simply describing people who come from parts of the world towards the East. What's so offensive about that? People routinely call Europe/America "the West" and nobody complains. Heck, Australia comes from the Latin word meaning "South" and they named their country that.
In fact, "Oriental" is just as imprecise as "Asian". Both need specification. Anywhere East of Europe/America is a huge expanse full of dozens of different peoples, just as calling someone "Asian" can refer to anyone from Iran to Japan.
Just to add further to the imprecise nature of using "Asian" to mean a very specific group of Asians, did you know that in the UK, "Asian" is typically used to refer to people from India and thereabouts? If you as an American tell an Englishman to meet you at the Asian restaurant down the street, you'll both find yourselves at different places.
All this to say, "East Asian" is the term that should be used for those who call themselves "Asian" in America. Using "Asian" alone is offensive to the billions in Asia who are not from the Eastern part of that continent.
The purpose of language is to convey meaning as precisely as possible. Appropriating a continent's denonym to refer to a subset of its inhabitants is purposefully muddying the issue.