The Cumberland Presbyterians are an entirely different denomination from the Presbyterian Church in America.
To make a very long story short, the PCA heritage is in the old southern Presbyterian Church in the United States or PC(US), as distinguished from the northern Presbyterian Church in the United States of America or PC(USA). Both the northern and southern churches merged together in 1983, after both had conservative secessions earlier on. The southern conservatives mostly left in the 1970s and 1980s to form the PCA; the northern conservatives left in the 1930s to form the Orthodox Presbyterian Church.
Rep. Akin’s seminary is actually from the RPCES which has a somewhat different and more complex background; that denomination merged with the PCA many years ago.
The Cumberlanders date back to a much earlier secession in the early 1800s by people who did not want to demand advanced seminary training for ministers on what was then the frontier (i.e., places like Kentucky and Tennessee) and were not convinced five-point Calvinists. The pastor who spoke with you has a point — the Christian Church (Disciples of Christ) is an heir of the Stone-Campbell movement which left several different denominations including Presbyterianism for similar reasons to the Cumberlanders.
A major shortage of ministers on the old frontier is largely responsible for the pressure to ordain men with little formal theological training. Obviously lots of highly educated men are in non-Presbyterian denominations, but it was this need to get more ministers that was responsible for the growth of Baptist, Methodists, and the Campbellite movement. Cumberlanders are people who shared many of the concerns from the same movement but didn't go as far.
Excellent short essay.
A lot of people have either forgotten the Stone-Campbell movement (and its place in the American Restoration movement) or swept it under the rug. The Campbellites are actually still a strong influence in many of the non denominational churches, although many of them don’t know it.