Shortly after World War II, in October of 1945, George Orwell wrote an essay titled "You And The Atomic Bomb", where he used the term Cold War to describe the coming nuclear weapons tensions he saw on the horizon, calling it a coming period of "peace that is no peace". Orwell did not specifically assign the term to describe a conflict between just the USA and the USSR at that time though.

When using the term Cold War to directly reference the specific post-war tensions between the USSR and the USA (and their satellite allies) though, that is widely credited to Bernard Baruch, who was a US political consultant, investor/financier and philanthropist. In April of 1947 in South Carolina, he gave a speech where he said “Let us not be deceived: we are today in the midst of a Cold War.” Later that same year, author/reporter Walter Lippmann really helped bring the term to prominence when he published a book about the subject that was aptly titled "The Cold War".

Was it simply called that because the conflicts were mostly fought by proxy?

Yes. To steal a quote from the wiki page on the subject, "It was termed as "cold" because there was no large-scale fighting directly between the two sides, although there were major regional wars, known as proxy wars...that the two sides supported."