I think your characterization that we are influenced by Calvin but not Calvinist is accurate. Of course it depends upon what you mean by Calvinist. Calvin wrote on lots of topics. But if you mean TULIP, I don’t think the Church as a whole is Calvinist, although there are Calvinists in the PCUSA.
Starting in 1903, with a modification to the Westminster Confession, the Presbyterian Church has backed away from strict Calvinism. In my opinion the 1903 change rejected at least limited atonement.
20th Cent confessional documents haven’t dealt with TULIP, so it’s difficult to be too clear on this, but my impression as a PCUSA elder is that many, maybe most, do not accept that God intentionally rejects anyone. I believe our Biblical scholars will tell you that this is the Biblical position as well.
That could be consistent with the late Luther’s position that justification comes entirely from God, but that why some people are rejected is a mystery, which we are better off not claiming to understand. The quote I give at the end of this posting from the Declaration of Faith seems to take a similar position. It says that we are saved only through grace, but is silent on what happens to others.
Here’s one of two treatments from the PCUSA web site:
http://www.presbyterianmission.org/ministries/today/predestination-2/. Note that it doesn’t give an answer on predestination, which is the key issue behind TULIP. It outlines 4 approaches. I think there are people within the PCUSA that accept each of them, and some that don’t feel that any of them is completely accurate.
I think the most complete description of PCUSA faith is the Declaration of Faith,
http://www.creeds.net/reformed/PCUSA1985/1985-int.htm. This doesn’t have confessional status, but was accepted by the General Assembly. Its advantage over the recent confessional documents is that it’s more complete. In general I commend it to your attention if you want to know the PCUSA position on major theological questions, although the Confession of 1967 is better the topics it covers.
Here’s the section that would be the most closely related to TULIP.
(3) The Spirit enables people to become believers.
The Spirit enabled people of all races, classes, and nations to accept the good news of what God had done in Christ, repent of their sins,
and enter the community of faith.
We testify that today this same Holy Spirit
makes us able to respond in faith to the gospel
and leads us into the Christian community.
The Spirit brings us out of death into life,
out of separation into fellowship.
The Spirit makes us aware of our sinfulness and need, moves us to abandon our old way of life,
persuades us to trust in Christ and adopt his way.
In all these things we are responsible for our decisions. But after we have trusted and repented
we recognize that the Spirit enabled us to hear and act. It is not our faith but God's grace in Jesus Christ
that justifies us and reconciles us to God.
Yet it is only by faith that we accept God's grace
and live by it.
