To, hopefully, make the issue of cossacks/Cossacks more clear, I will allow myself short quote from my book

"Before we look into detailed description of this formation, we need to first of all clarify some misunderstanding in regards to its name: both in Polish and in English. Originally this cavalry was called
jazda kozacka (cossacks cavalry), with its soldiers being known as
kozacy (written with lower case). It was term used to describe free man, ‘serving for money’ and was adopted for both military use and to name household servants, often mounted. In 16th century word started to be used to describe free men settled in Ukraine settled near Dniepr river but they were called
Kozacy (written with upper case). As such, in original Polish text one could find both
kozacy and
Kozacy serving together in one army. Obviously it made it very confusing when translating it into English, when normally always one form – Cossacks – is being used, leading to many errors in identifying who is in fact described in the text and attributing many military features of Polish cossacks cavalry to Zaporozhian Cossacks. As such, in this book we will be using term cossacks cavalry or cossacks to describe regular cavalry in Polish army and Cossacks to describe Zaporozhian Cossacks"