Moderate Calvinism strives for the Biblical balance in all things. It does not seek a mid-point between Arminianism and Calvinism (I doubt there is such a thing), but a balance of all the key components in Reformed soteriology. In terminology, moderate Calvinists range from total rejection of limited atonement (sometimes called "4-point Calvinism") to full acceptance of limited atonement with the addition of other balancing factors. Well-known Calvinists who reject limited atonement would include Bruce Ware and Randy Alcorn (a historical representative would be Donald Grey Barnhouse).
In my study, most Calvinists (including quite a few high Calvinists) take a position somewhere between the view that strictly limits every benefit of the atonement to the elect and an outright denial of limited atonement. There are lots of ways to articulate a middle position on the extent of the atonement, and this seems to be the course followed by most Reformed thinkers, whether they are classified as moderate or high, 4-point or 5-point, 4.25, 4.5, 4.75, or whatever. There can be a lot of needless hair splitting, mislabeling and debating about this, but I respect any position that is orthodox, sincerely held, and thoroughly grounded in Scripture. Folks who are just toeing the party line and repeating the philosophical conclusions logically demanded by their systematic theology aren't adding anything to the discussion, as far as I'm concerned.
I find that moderate Calvinists whose church polity requires allegiance to all 5 points (e.g., Presbyterians) tend to qualify "limited atonement" as being both limited and unlimited, but in different senses. The classic formulation is "sufficient for all, but efficient for the elect only." I observe that moderate Calvinists whose church polity does not require allegiance to all 5 points tend to call themselves 4-point Calvinists and reject limited atonement altogether (at least in their terminology). There are, of course, many exceptions, but the point is this: approaches to the atonement that are very nearly the same can be called 4-point or 5-point Calvinism interchangeably - depending on the context.
John Piper describes the moderate position well when he says this:
We do not deny that all men are the intended beneficiaries of the cross in some sense. 1 Timothy 4:10 says that Christ is "the Savior of all men, especially of those who believe." What we deny is that all men are intended as the beneficiaries of the death of Christ in the same way. All of God's mercy toward unbelievers—from the rising sun (Matthew 5:45) to the worldwide preaching of the gospel (John 3:16)—is made possible because of the cross. (SOURCE)
Jack Brooks describes it in a slightly different way, but with his characteristic candor and wisdom, here:
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I was recently asked why I described myself as a moderate Calvinist. The "moderate" part reflects the fact that I understand the Scripture to teach that, in some sense, Christ died for everyone without exception. Christ provided the possibility of redemption for everyone, and an actual redemption is applied only to the elect -- those who are given the gift of faith. The logic problems that my five-point Calvinist brethren immediately raise don't sway me, since the only concern I choose to have is over what specific verses actually say in the original languages -- not how many logic syllogisms my view might seem to contradict, or how one escapes "double jeopardy", and so on. Those philosophical objections don't matter to the question, "what does this verse mean?" because, in the end, they are philosophical objections, not exegetical questions. Only exegetical principles matter when one is asking the question, "What does this verse mean?" The question, "How can I reconcile this with these other ideas over here" is a secondary, or even a tertiary, concern. Not a primary concern. Limited atonement does not survive consistent, thorough-going exegetical analysis. It's our job to iron out any wrinkles that the exegesis might create in the over-all fabric of our systematic theology.