I would call Skyclad the original folk metal band. They took speed/thrash metal and infused it with violins and a traditional, nearly medieval sense of melody. That's basically what Korpiklaani is doing, except using finnish folk music and going overboard with the instrumentation.
For me, Viking metal and Folk metal are two different entities. Pagan metal is iffy, but I usually use the 'Pagan' tag for bands playing folk metal influenced by traditional slavic themes and folk playing. Slavland, Pagan Reign, Arkona, Butterfly Temple, Skyforger, etc...
As far as viking and folk metal goes, the primary difference I look for are traditional folk instruments. All bands dubbed 'folk metal' have traditional folk instrumentation, or a keyboard imitating folks instruments during the songs. Viking metal forgoes this and focuses more on a guitar riff attack--without sacrificing the battlefeild atmosphere. Look at Einherjer for example. They used keyboards, but only for string synths and pads as opposed to flutes and such. The basis on Einherjer songs lie in the guitar riffing. The riffs aren't super melodic, but you can't deny that there is a heavy amount of scandinavian folk influence there. One could also talk about the influence on viking metal compared to folk metal, but I don't want to make this post any more long-winded, haha.