Are there distinctions between the various terms for "Confucius" (孔子, 孔丘, 仲尼)?
First you need to know about how people were named at that time. Back then one's name had three parts:

姓 (family name), 名 (given name), 字(also given name, but doesn't exist today)

孔 was his 姓, 丘 was his 名, and 仲尼 was his 字.

I don't need to explain the family name much. As for 名 and 字, the difference is that 字 is more formal than 名. And one often calls himself by 名 to show modesty and calls the one he is talking to or mentioning by 字 to show respect.

Finally, "子" is a suffix after the family name to mention someone who is well-respected (for example, a teacher, a well-educated person, etc., and Confucius was both and more).