Why do some people go by First Initial, First Name, Last Name?

I’m reading a book by W. Michael Gear and Kathleen O’Neal Gear.
That set me to thinking: I know of many famous people who go by Initial(s) + Last name (e.g. J.J. Abrams) and many who go by First Name + Middle initial + Last name (e.g. Samuel L. Jackson, Michael J. Fox).
I also know that many people use their middle names rather than their first names.

But what’s up with the use of a first initial and middle name? Why not just get rid of the first initial altogether?
The only other example I can think of right now is J. Michael Straczynski, a comic writer and general creator of nerdular nerdence.