I think the Alliance side-steps the issue. I think they are also cautious about labels in part because many of them evolve (for example what it means to be a Democrat today is not what it meant 70 years ago, similarly "Methodist" rarely means John Wesley's theology, and we all know about "gay"). Furthermore, as was mentioned there is a lot of confusion about what "Calvinism" really is. For many it is simply a word for "predestination" and again as you said for many that means "we don't need to go and preach the Gospel".
Anyway I wanted to quote from the first chapter of "The Pursuit of God" so I typed in a few words I remembered and I found it. The interesting thing is that where I happened to find it was on a page for the "Society of Evangelical Arminians". Many of the Calvinists I read and listen to quote Tozer often...maybe the Arminians do as well... Anyway below is the opening of "The Pursuit of God".
Christian theology teaches the doctrine of prevenient grace, which briefly stated means this, that before a man can seek God, God must first have sought the man.
Before a sinful man can think a right thought of God, there must have been a work of enlightenment done within him; imperfect it may be, but a true work nonetheless, and the secret cause of all desiring and seeking and praying which may follow.
We pursue God because, and only because, He has first put an urge within us that spurs us to the pursuit. “No man can come to me,” said our Lord, “except the Father which hath sent me draw him,” and it is by this very prevenient drawing that God takes from us every vestige of credit for the act of coming. The impulse to pursue God originates with God, but the outworking of that impulse is our following hard after Him; and all the time we are pursuing Him we are already in His hand: “Thy right hand upholdeth me.”
In this divine “upholding” and human “following” there is no contradiction. All is of God, for as von Hegel teaches, God is always previous. In practice, however, (that is, where God’s previous working meets man’s present response) man must pursue God. On our part there must be positive reciprocation if this secret drawing of God is to eventuate in identifiable experience of the Divine. In the warm language of personal feeling this is stated in the Forty-second Psalm: “As the hart panteth after the water brooks, so panteth my soul after thee, O God. My soul thirsteth for God, for the living God: when shall I come and appear before God?” This is deep calling unto deep, and the longing heart will understand it.