This is a partial answer. It is only a suggestion of where one might start looking for answers to the following questions.
Does anyone know of thought experiments in the philosophical literature that illustrate essential features of free will?
For an unusual approach to free will see John Conway and Simon Kochen's Free Will Theorem. What makes this unusual is that they claim to have shown that if we have free will then so does a quantum system.
What makes that striking is that free will is often associated with being rational, however, a quantum system does not appear to have the ability to be rational. If this is true then rationality is not an essential feature of free will.
Does anyone know of a philosophical work which offers an exposition of the concept of free will?
Outside of querying the Stanford Encyclopedia of Philosophy, the Internet Encyclopedia of Philosophy or Wikipedia, the collection of articles edited by Derk Pereboom called Free Will in the Hackett Readings in Philosophy provide a collection of primary sources on free will including Aristotle, Augustine, Spinoza, Hume, Kant, Reid, van Inwagen, Kane and others.
Mark Balaguer's Free Will is an easy to read introduction to the physicalist perspective on free will. Even if you are not a physicalist it is worth reading for its clear exposition.
Are there different (named) conceptions of free will in the philosophical literature so far as anybody knows?
One place to go for a quick overview is Wikipedia's "Free will" article.
Balaguer, M. (2014). Free will. MIT Press.
Conway, J., & Kochen, S. (2006). The free will theorem. Foundations of Physics, 36(10), 1441-1473.
Pereboom, D. (Ed.). (2009). Free will. Hackett Publishing.
Wikipedia, "Free will", https://en.wikipedia.org/wiki/Free_will