While ‘com-’ refers to ‘together’ and ‘-panis’ refers to ‘bread’ in Latin, ‘companio’ evolved into ‘company’ later on. Each country has its own classification for companies. In fact, even the use of ‘company’ and ‘cooperation’ has different timing. The British tend to use ‘company’ more often, and the American tend to use ‘cooperation’! 

Any company is suitable for .company! You can include what you do in your website to let searchers know at first sight!