The term "Star Wars" was not coined by Reagan or his staff but rather the program's critics. As far as I know, Reagan never used the term and wished journalists would stop using it because calling it Star Wars implied (correctly) that the program was based more on science-fiction fantasy than actual science.

For example, here's an exchange between a journalist and Reagan during a press conference Q&A:

Q. Mr. President, if you are flexible, are you willing to trade off research on "Star Wars'' technology for deep cuts in the Soviet nuclear arsenal, or are you set in concrete, as your advisers say, against any negotiations on "Star Wars''?

The President. Well, let me say, what has been called ``Star Wars'' -- and, Helen, I wish whoever coined that expression would take it back again -- --

Q. Well, Strategic Defense -- --

The President. -- -- because it gives a false impression of what it is we're talking about...

Source: Reagan Presidential Library, Public Papers of the President