Are there any other agreed-upon definitions of "free will" within mainstream Christianity?

In the Thomist tradition, "The will is a rational appetite." (ST I-II q. 8, a. 1, co.)

As for the definition of appetite:

To desire or have appetency (appetere) is nothing else but to strive for something (ad aliquid petere), stretching, as it were, toward something which is destined for oneself. - Quæst. disputatæ, De veritate, Q. 22, a. 1

A tendency, an inclination, or direction. - The Catholic Encyclopedia

The "free" in "free will" follows from the very definition of will, so "free will" doesn't need to be defined separately. Here's a summarized explanation why.

The rational appetite is contrasted with the sensitive appetite in that the rational appetite follows (is inclined to) intellectual knowledge while the sensitive appetite follows sense-cognition. Sense-cognition apprehends only particulars, but intellectual knowledge apprehends universals. It follows from this detail that the will is intrinsically free, because by following "good as such" it is not bound to follow any particular good:

But a rational nature, being closest to God, not merely, like inanimate things, has an inclination to something, and, like a sentient nature, a mover of this inclination determined as it were extrinsically, but further so has its inclination within its own power that it does not necessarily incline to anything appetible which is apprehended, but can incline or not incline. And so its inclination is not determined for it by anything else but by itself. This belongs to it inasmuch as it does not use a bodily organ; and so, getting farther away from the nature of what is moved, it approaches that of what moves and acts. It can come about that something determines for itself its inclination to an end only if it knows the end and the bearing of the end upon the means to it. But this belongs to reason alone. Thus such an appetite, which is not determined of necessity by something else, follows the apprehension of reason. - Quæst. disputatæ, De veritate, Q. 22, a. 4

In the natural and the sensitive appetites there is no freedom. One is necessitated by the laws of nature itself, the other by the sense-apprehension of a concrete thing as pleasant and useful. The will, on the contrary, is not necessitated by any concrete good, because no concrete good fully realizes the concept of perfect goodness which alone can necessarily draw the will. In this is to be found the fundamental reason of the freedom of the will. - The Catholic Encyclopedia

It's a succinct and robust definition but with a somewhat large metaphysical import, as it depends on intrinsic teleology and scholastic jargon. Here "appetite" is being used not simply as a psychological phenomenon but as something "found in all beings, even in those that are unconscious" (CE), being basically a shorthand for final causes, which are amply denied in modern thought. It is not Libertarian because it posits that there's still something which no will can possibly override: its own inclination towards good. So, even while choosing evil, it is still choosing something under the aspect of good (cf. ST I-II q. 8, a. 1, ad 2). This also puts it in opposition to the Reformed conception, but that's another discussion. All that is said here is independent of whether we're dealing with the will of fallen man or man in a state of grace, or even whether we're dealing with human will (much also applies to angelic and divine will; beasts have no will).

The debate between Erasmus and Luther is one of the earliest of the Reformation over the issue of free will and predestination, which is a form of determinism.

Erasmus wrote "On Free Will" in September of 1524 as his first public attack upon Martin Luther. While he believed that the Roman Catholic Church needed reform from within, he accused Luther of having gone too far. He held that every human possessed free will and that the doctrine of predestination stood at odds with the teachings of the Bible.

Erasmus argued against the belief that God's foreknowledge of events caused those events, and he held that the doctrines of repentance, baptism, and conversion depended on the existence of free will. He likewise contended that divine grace called, led, and assisted humans in coming to the knowledge of God, and then supported them as they used their free will to make choices between good and evil, and enabled them to act on their choices for repentance and good, which in turn could lead to salvation through the atonement of Jesus Christ. - Wikipedia

In response in December 1525, Martin Luther published "On the Bondage of the Will" or "Concerning Bound Choice".

Luther's response was to reason that original sin incapacitates human beings from working out their own salvation, and that they are completely incapable of bringing themselves to God. As such, there is no free will for humanity because any will they might have is overwhelmed by the influence of sin. Luther concluded that unredeemed human beings are dominated by obstructions; Satan, as the prince of the mortal world, never lets go of what he considers his own unless he is overpowered by a stronger power, i.e. God. When God redeems a person, he redeems the entire person, including the will, which then is liberated to serve God. - Wikipedia

I think these two (and most) positions on the extent of the human will's freedom acknowledge what might be called Free Agency; that is, the ability to choose from within a limited set of options. Libertarian Free Will, for example, easily admits that one is not free to choose to levitate or grow an additional set of eyes, etc., and that what one is free to choose are only those possible options available.

Other positions, like the Reformed stance, would agree but would severely limit the scope of what is possible by extending the limitation into our very natures:

“Free agency is a general mark of all human beings as such. All humans are free agents in the sense that they make their own decisions as to what they will do, choosing as they please in the light of their sense of right and wrong and the inclinations they feel. Thus they are moral agents, answerable to God and each other for their voluntary choices. So was Adam, both before and after he sinned; so are we now, and so are the glorified saints who are confirmed in grace in such a sense that they no longer have it in them to sin. Inability to sin will be one of the delights and glories of heaven, but it will not terminate anyone’s humanness; glorified saints will still make choices in accordance with their nature, and those choices will not be any the less the product of human free agency just because they will always be good and right.” - J.I.Packer

Packer's position (and that of Johnathan Edwards as well) is that the factors limiting choice are internal, i.e., our very natures corrupted to the point of an unavailability of the option to actually choose to do good. What we do choose, however, are "free" choices from the set of choices available.

Thus, the Free Agency view can apply across the board and the distinction in various frameworks boils down to describing of what the limitation of possible choices consists. In other words, is exercising moral free agency in choosing to do good an available option for everyone regardless of salvation, an available option tending toward salvation, or the product of salvation.

Donald Mackay contributed a view of free will that he called "Logical Indeterminacy". The central point was that there is no prediction of what Fred will choose next that, if presented to Fred as a prediction, he would be logically compelled to accept as correct. The reason is that the very act of presenting this prediction to him would change his state of mind such that the prediction itself would be invalidated.

His original paper seems to be available only behind various pay walls, but there is a freely-available article from Historical Theology - The contribution of Donald Mackay that provides a good summary.

I've not seen this view very widely discussed, and it is perhaps questionable whether it provides enough 'freedom' to statisfy the average man in the street's sense of what free will ought to be.

If Libertarian Free Will means human wills are able to make decisions without any external compelling force, then there are a variety of Christian views which would claim they teach it, ranging from Open Theism, to Arminianism's prevenient grace freeing wills that were bound, to teaching that only Adam & Eve had LFW, to possibly some forms of compatibilism that our wills are free in the LFW sense, but how they're compatible with God's total sovereign will is an absolute unknowable mystery. So simply saying LFW won't be enough to pin down exactly what someone believes.

But the main alternative understanding of free will that I'm familiar with is the Reformed version. (Catholicism likely has a different view again.) The Reformed understanding is a compatibilist view in which the creation design of man has a will which is free to choose to do good or evil, but in sin we have lost the ability to will to do good. When God saves and redeems a person they are freed from the bondage of sin so that they can will to do good again.

The Westminster Confession sets forth this understanding in chapter 9:

1. God hath endued the will of man with that natural liberty, that is neither forced, nor by any absolute necessity of nature determined to good or evil.

2. Man, in his state of innocency, had freedom and power to will and to do that which is good and well-pleasing to God; but yet mutably, so that he might fall from it.

3. Man, by his fall into a state of sin, hath wholly lost all ability of will to any spiritual good accompanying salvation; so as a natural man, being altogether averse from that good, and dead in sin, is not able, by his own strength, to convert himself, or to prepare himself thereunto.

4. When God converts a sinner and translates him into the state of grace, he freeth him from his natural bondage under sin, and, by his grace alone, enables him freely to will and to do that which is spiritually good; yet so as that, by reason of his remaining corruption, he doth not perfectly, nor only, will that which is good, but doth also will that which is evil.

5. The will of man is made perfectly and immutable free to good alone, in the state of glory only.

The advantage of this view is that it accounts for the Biblical texts using "bondage of sin" language, as well as the universal experience of the Christian that despite us wanting and choosing not to sin, we still do (see Romans 7). If LFW proponents aren't just saying that free will existed only for Adam and Eve before the Fall then these things are harder to account for.

I've heard this view explained as freedom being about living as you were designed to be. Roger Olsen writes:

As far back as Augustine in the late fourth and early fifth centuries, some Christians have defined free will simply as “doing what you want to do” so that you are acting freely whenever you are not coerced to act against your will. Or, as the great Puritan theologian Jonathan Edwards argued, you act freely whenever you act according to your strongest motive that you do whenever you act without external compulsion.

Rather than LFW where freedom is about being free from anything which would bind the will, the Reformed view focuses on freedom to live for Christ as we serve him and love others. When God breathes life into spiritually dead hearts of stone is what matters that we have the ability to choose death again, or that the desires which shape our wills are now inclined to God? Of course Reformed writers do speak about both directions of freedom (for example, this Desiring God article.)

The disadvantage is that this version of free will is unusual, whereas the LFW understanding is the dominant one in both philosophy and common usage. But the freedom to concept does connect to the philosophical topic of teleology. To only be truly free when we live as God intends runs completely counter to the dominant Western priority of self-determination, but then, self-determination can be seen as one of the foundational sinful inclinations.