I am not a native speaker and terrible at googling this, but I've once heard that 電腦 was originally the Taiwanese way of saying it and 計算機 the Chinese way, in the same way that 'program' is still called 程式 on one side of the strait and 程序 on the other, or 'internet' is either 網絡 or 網路. The names of movies and video games are also generally different (PRC, HK, TW).

計算機 is a calculator, not a computer, in Taiwan: https://tw.images.search.yahoo.com/search/images?p=計算機

Compare with Baidu in the PRC: http://image.baidu.com/search/index?tn=baiduimage&ie=utf-8&w...

Edit: Oops, I didn't even realise that the parent poster asked about 計算著. That one I've never seen.

计算着 [jisuanji] is the older usage that harkens back to the days when computers were mainframes and terminals and not in every household. Generally everyone says 电脑 [diannao] these days.

计算机 jisuanji would be a hardware computer, 计算着 jisuanzhe computing as an ongoing action, and 计算者 jisuanzhe computer in the sense of one who computes

No clue. I did a copy and paste.