There is so much heat generated with this topic - you'd think it was the most important doctrine of all.
So, like the OP title asks - who thinks that it matters eternally what one believes about these positions?
A good, but difficult question. There are boundaries of belief content beyond which I--well, probably most of us--would say a person is not a Christian, usually willingly so on the part of the non-Christian. I think in particular of the tests of faith in 1 John such as confessing that Jesus came in the flesh and is the Son of God.
And then there are doctrines about which I think most professing Christians would say salvation does not depend--millenial position or certain views of baptism come to mind.
Doctrines touching on salvation and the nature of God as it relates to salvation is a touchy and difficult middle between these above sides, or so it seems to me.
My hope is that there are many whose doctrine of salvation is heterodox to some degree but who are saved nonetheless, in part I think because salvation is existential, experiential, and accomplished (for the moment let us say synergistically or monergistically) by God. And surely there are those in one soteriological camp or another confessionally who are not saved.
This is not to say the debates are unimportant or unimportant for salvation; salvation is rather important and depends in part on what is believed, content which is important to the God who saves in Jesus Christ.
What is also important, I think the apostolic record would suggest for example, is the moral character evident in the way the salvation doctrine is argued--contending for the faith in uncontentious manner, if you will. Being gentle and humble in one's argument, even if at times forceful. In the absence of the fruits of salvation or where impatience or disrespect is evident in the argument, the manner of argument contradicts something about the salvation doctrine ostensibly argued.
P.S. Helpful as OzSpen's summary chart "Calvinism, Lutheranism, Arminianism, 'The Rest of Us'" may be, I do not think it accurately or fairly represents Calvinism at least as I see it, to speak as a Calvinist. But that may be another story. Others from other positions may weigh in about the representations of their positions too.