I think that Methodists also believe that theirs is the only true Church.
This is totally 100% wrong. My father was a Methodist before he met my mother, and I've known MANY Methodists through the years. They are one of the most open, ecumenical groups among those that profess Christian doctrine yet have (in some parts) remained among the more conservative groups (still holding to biblical authority).
My church history prof had a funny saying - "You need the Baptists to get them saved, the Presbyterians to get them theologically astute, and the Methodists to give them a social conscience." Rigidity is found among the Churches of Christ that are proliferated through the South, with major numbers in Tennessee and Texas. There are SOME Baptist groups (Landmarks is one example) that are extremely separatist and militantly fundamentalistic although you won't find those folks (usually) going all Al Qaeda on you.
Here's how I look at it (not that anyone asked): the centrality of NT teaching begins with the bodily Resurrection of Jesus Christ. Our theology should go in concentric circles outward from there. Most Christians who apostasize and abandon the faith have set their theology up like a domino - when one falls, the entire system collapses. If one sets up his bibliology first (doctrine of the Bible - inerrancy) and is then persuaded the Bible has errors then the entire worldview of such a person collapses. (Amazingly enough, fundamentalists who do this don't usually become more sophisticated in their approach - they become militant skeptics who are every bit as fundamentalist as they were before but just play for the other team - just as angry and snide as they were as professing Christians).
Let me put it another way: belief in young earth Creationism (which is NOT a biblical teaching but a philosophical construct) is NOT even on the same planet (if you'll pardon the pun) as the bodily Resurrection of Jesus. In fact, if you will look back there are Christians arguing for a much older than 6,000-year old earth hundreds of years prior to Darwin. The gap theory dates to the 17th century (the notion there is a creative gap between Genesis 1:1 and 1:2). I'm not going to argue the correctness or incorrectness of the theory except to say one can be a devout, orthodox Christian and still believe in an earth MUCH older than 6,000 years.
I just don't believe a Supreme Being is under any obligation to make a person use the brain He gave them. By the same token ALL positions including atheism mandate elements of faith (none of us knows for absolute certainty what will happen after we die - until we die).
But back to the point - Methodism historically has been an ecumenical movement, open with a social conscience. I don't agree with some of their positions but I've had more than one person suggest I "convert." I can't, though - I have too many disagreements on some "lesser" issues with them to "fake it."
Catholicism is also nowhere near as closed now as it was pre-Vatican II.