Jonathan Israel's Radical Enlightenment Jonathan Israel uses the term "Radical" (in the context of the Enlightenment, of course) to describe a set of overlapping (but not necessarily unconditional) ideas pertaining to: God (e.g. religious fundamentalism vs. atheism), political philosophy (e.g. absolute monarchy vs. democratic republic) and ethics (e.g. anti-tolerance vs. universal tolerance). He contrasts the Radical arm of the Enlightenment with the Moderate arm, and the Counter-Enlightenment arm.
Israel doesn't provide an exact, strict , or inflexible criteria for what constitutes a "radical" Enlightenment thinker, as philosophers during this time differed in the above views, sometimes inconsistently, and across a spectrum. Rather, Israel focuses on a multitude of philosophers, discusses their views within the socio-political and intellectual environment they lived and worked in, and argues where they fit within the different strains of Enlightenment thinking, while acknowledging that one can hold views from different "sections" of Enlightenment thought. For example, David Hume, while irreligious, and arguably an atheist, nevertheless believed that organized religion was necessary for society to properly function. And despite his somewhat advanced ethical views, he was nevertheless highly skeptical of democracy, preferring a mixed monarchy. Compare this with Diderot, who emphatically advocated for atheism, democracy, and universal ethics. In fact, while Diderot and Hume wrote to one another occasionally, they never really took the others' idea seriously. As such, Israel argues that Hume should be more accurately viewed as a member of the Moderate Enlightenment, rather than the Radical arm with Diderot, D'Holbach, (early) Helvetius, or, of course, Spinoza.
Israel actually devotes quite a few pages to Jefferson, who, save for Thomas Paine, is arguably the most radical of the Founding Fathers. While Jefferson was undeniably influenced by Locke (who was certainly not a Radical), much of the language of the Constitution is written in "Radical" terms (e.g. "All Men Are Created Equal" could never have been derived from Locke). In regards to Price, Priestly, and Wollstonecraft, while they did believe in a providential universe, that fact alone doesn't exclude them from holding highly radical ideas pertaining to, e.g. political philosophy, and ethics, such as women's emancipation. One can certainly be a Deist and believe in universal toleration, or democracy, both of which were certainly radical during the Enlightenment.
The Radical, Moderate, and Counter-Enlightenment placards are really just helpful guidelines in order to compare and offer resemblances, influences, etc. between philosophers and theologians during the Enlightenment. There is no rigid demarcation for who fits in where, as that would be messy and ultimately ahistorical.