Any criticism of Lincoln as unorthodox also applies to many or most of the leading lights of America's founding generation. Indeed, in some ways, Lincoln was more religious than Washington, Adams, or Jefferson. The age of enlightenment had passed, and been replaced by a more evangelical temper.
For Lincoln and many in his day, individual reading of the Bible and attempts to understand and apply such truths as it contained took precedence over church membership and dogma. America embraced a Protestant notion of individual conscience and judgment in matters of religion, and it's too late to overturn that tradition or to fault those who followed it, rather than rely on dogma.
Lincoln did regularly attend Presbyterian services after 1850, but declined to become an official member of the church. The difference between Lincoln's generation and that of the founders is that in 18th century Massachusetts or Virginia and 19th century Illinois one would automatically be assumed to be a member of the established church in colonies with state-supported churches unless one explicitly denied membership. In the next century, on the moving frontier, church membership was often assumed to be a matter of inspiration and revivalism, and people who would have been passively counted as Anglicans or Congregationalists in colonial times remained unchurched because they didn't feel a calling. It's estimated that only 23% of Americans were church members in 1860.
It ought to be clear that if Lincoln believed in Providence, in some Almighty or Most High who affected the course of events on earth he could not be an atheist, agnostic, or deist. Still less was he a Muslim, Hindu, or Animist. Part of the confusion stems from Herndon, who seems to have been the sort of fellow who assumes that if you share an office, drink, or joke with him that you automatically share all his opinions about religion or politics. Herndon doesn't take the time to question, analyse, or try to understand beliefs that aren't his own.
And part comes from the fact that these neo-confederates don't really take categories and distinctions seriously. If they approve of someone's political point of view they will forgive any amount of doubt or heterodoxy. If they don't, they'll make all divergences of opinion or independence of mind into heresy. The idea seems to be to formulate an indictment in which every accusation made against Lincoln becomes another charge or count, rather than to come up with a picture of a complex human being that puts all his characteristics in place.
So much of this League of the South claptrap shouldn't be taken seriously. And indeed, on the eve of one of the most important elections of our lifetime, such third-rate Lincoln-bashing is pretty unconscionable. Those who are truly interested in Lincoln's religious beliefs will consult some of the fine serious works published in recent years by Allen Guelzo, Stewart Winger, Richard Carwardine, or Lucas Morel and not bother with all the stupid trash talk.
Mark Noll has a fine article on Lincoln and religion here: "The greatest difficulty in coming to a clearer picture of Lincoln's faith is the fact that his religion does not fit into modern categories. He was not an orthodox, evangelical, "born-again" Christian striving toward the "higher life" (as these terms have been used since the 1870s). But neither was he a skeptical "modernist" with a prejudice against the supernatural and an aversion to the Bible."
The question of what Lincoln believed is a complicated one. Why bother with those who don't want to give the matter serious thought or consideration? The difficulties involved in knowing or understanding anything of theology or history are such that there's little point in wasting one's time on fanatics, sophists, and propagandists.
Lincoln's religious beliefs weren't so very different from Washington's, and Washington has also been criticized for being unwilling to make much mention of Jesus Christ in his writings and utterances.
While it is true that Washington was not particularly known for his public expression of religion and may have bordered on more formalized forms of deism at times in his life, what we do know about his practice is indeed substantially different from Lincoln. Whereas Lincoln never belonged to any formal church and never professed any affiliation with Christianity, Washington was a practicing High Episcopalian. He was a vestryman in Fairfax Parish at the Falls Church for several years and was appointed Church Warden in 1763 with the task of contracting the construction of a new chapel. After the parish split in the 1760's he again served as an Episcopal vestryman in the new Truro Parish, encompassing Mount Vernon. What is known of Washington's concept of god indicates that his God was the God of Abraham, as he wrote, even in some of his more "deist-like" letters. Lincoln, by contrast, tended not to identify his concept of god with the judeo-christian one.
Any criticism of Lincoln as unorthodox also applies to many or most of the leading lights of America's founding generation.
You are simply incorrect about that. While it has been said that Jefferson's religion is a great mystery (the common quote is that Jefferson's beliefs are "known only to himself and God") and a few others dabbed in unitarianism (i.e. Adams) and deism, the overwhelming majority of them were practicing members of an organized denomination. Of the signers of the Declaration of Independence the only persons without a mainstream Christian religious affiliation were Jefferson, as previously mentioned, Franklin - a deist, and unitarian Adams. Just about everybody else fell into one of five different mainstream denominations: episcopalian, presbyterian, congregationalist, baptist, or roman catholic.