This video link will probably disappear sooner rather than later but in the meantime it's an interesting look at one variety of HEMA (Historical European Martial Arts). These people are inspired by the techniques of the late medieval German masters, but though they strive for historical authenticity in their moves they are not interested in other kinds historical re-enactment or re-creation. No "thees and thous," as one of them puts it. (But note that there is one guy on the sidelines who seems to be wearing some kind of medieval clothing and maybe a coronet.)
There is a wide variety of martial arts out there these days, different organizations pursuing different goals. This may lead to the Balkanization of a field of activity once completely dominated by the Society for Creative Anachronism.