What is the Reformed Protestant understanding of God's Permissive Will, His Sovereign or Decretive Will and Efficacious Will?

Your quote source is part of R.C. Sproul's explanation of Permissive Will, so you will find the context by reading the complete article titled "Discerning God’s Will: The Three Wills of God". After a quick and relatively straightforward definitions of Decretive and Prescriptive wills, he then warns us of "the faulty distinction between willing and permitting" by quoting Calvin along with the Biblical support Calvin used for his thesis. Sproul then ends with a quote from St. Augustine.

A related resource I found is a free book by R.C. Sproul "Can I Know God's Will" which I could also find the PDF here where he also discussed Decretive and Preceptive Will of God but introduced "God's Will of Disposition" as roughly equivalent to his "permissive" will, similar to how this article also divide God's will into the same 3 categories. That's in chapter 1 ("The Meaning of God's Will") to which he adds chapter 2 ("The Meaning of Man's Will") followed by 2 application chapters on job and marriage.

Other related resources I think you'll find valuable are:

1. Nathaniel's answer to a related question which gives you the relevant Biblical verses for the Decretive and Preceptive Will of God based on R.C. Sproul and Charles Hodge

2. Charles Hodge's treatment of The Will of God in his famous (although dated) 1871 Systematic Theology Volume 1 Chapter V Section 9

3. Blog article "A Theology Of the Will of God by Adam Setser, a Southern Baptist theology student, surveying different ways (other than decretive vs. preceptive) to view the two sides of the will of God within the Reformed tradition, nicely paraphrasing some of Hodge's treatment in 2015 English.

I found Adam Setser's treatment to be the most helpful to address your confusion because he goes into the Latin words of 5 different pairings of tensions that Reformed theologians have identified over the centuries. I put in bold the terms you are including in your question. From the analysis below, I think the confusion comes from mixing terms from different pairings.

1. The backbone: voluntas beneplaciti (will of desire) / voluntas decreti (will of decree) / voluntas occulta (secret will) VS. voluntas signi (signifying will) / voluntas revelata (revealed will) / voluntas praecepti (will of precept) (see subsection C in Hodge)

2. Choice: voluntas absoluta VS. voluntas conditionalis (see subsection E in Hodge)

3. Causation: voluntas antecedens VS. voluntas consequens (see subsection D in Hodge)

4. Theodicy: voluntas efficiens (efficient will) VS. voluntas permittens (permitted will)

   If God’s will must account for all creation, what about evil, Satan, and hell?

   Voluntas efficiens: the “efficient will” refers to those aspects of His will who receive His full affirmation. Efficiens also means to create, cause, or produce. So this is the product of His creation which obeys Him perfectly and gets his affirmation.

   Voluntas permittens: the “permitted will” refers to the part of creation which doesn’t get His affirmation. He still has willed its existence, but He isn’t pleased with it. He permits it to exist because it serves His purposes (e.g., Satan).

5. Efficacy: voluntas efficax (effective will, efficacious) VS. *voluntas inefficax** (ineffective, HERETICAL!)

If the above terms are not enough, there are a few more distinctions in the famous Reformed Systematic Theology textbook (1938) by Louis Berkhof: Part One (Doctrine of God), THE BEING OF GOD Chapter VII (The Communicable Attributes), Section D (Attributes of Sovereignty), pages 82-87, HTML version here, PDF here. These 6 pages I think will be the best resource to integrate all the above distinctions into a coherent whole within the Reformed tradition where Dr. Berkhof said how the antecedent/consequent and absolute/condition pairings found little favor in Reformed theology while the decretive/preceptive, eudokia/eurestia, beneplacitum/signum, and secret/revealed pairings were more generally accepted. He discusses the issue of "permissive will" in a subsection titled "God's will in relation to sin". Finally by necessity of its being a textbook Dr. Berkhof's treatment also adds contrast between Reformed understanding of the sovereign power of God to some modern non-Reformed positions (Strauss, Schleiermacher).

While the above resources are "certifiably" Reformed in the theology (despite different theologians, e.g. Dr. Berkhof and Dr. Hodge, organizing the discussion of God's will differently), I think the following article titled "Providence of God" from the 1997 Baker's Evangelical Dictionary of Theology should provide you with a larger context to place the will of God in a broader as well as more ancient perspective, contrasting how in OT, 2nd Temple, and NT traditions Jewish and Christian believers view God's will in PERSONAL terms rather than IMPERSONAL fate / deterministic way that the Greco-Roman philosophy adherents (such as Stoics) view God's will.

One way to consider God's will in an even larger context is of course by studying Louis Berkhof's textbook treatment of THE WORKS OF GOD where he begins by discussing The Divine Decrees in General (Chapter I), Predestination (Chapter II), Creation (Chapters III to V), and ends with Providence (Chapter VI).