"Axel Jump" in jazz and figure skating - question about etymology

Figure skating is older than Jazz Dance as a discipline. Although the African origins of Jazz Dance are ancient it did not become a recognised dance form with named moves and choreography until the early 20th century when white society took it up and added white styles to it. On the other hand Brittanica says that the first book on figure skating was published in 1772. Obviously this would not have been anything like modern figure skating but the Brittanica entry also mentions the revolutionary contribution of Jackson Haines based on ballet and formal social dance to figure skating in the 1860s.

Given this history it can be seen that figure skating was well established when Axel Paulsen was born in 1855 and he was both a speed skater and a figure skater. Also Haines had greater success with his freer style in continental Europe than his native US so would have influenced Paulsen.

Again according to the Brittanica entry Paulsen first demonstrated the turn named after him in 1882 at the first international competition

The development of both modern figure skating and modern jazz dance took place in the early 20th century but by then the skating world was already aware of the Axel Turn and, of course, its inventor (who died in the 1930s). Jazz Dance must have adapted the Axel Turn from figure skating, not the other way around. Remember that the jazz dance version is different from the figure skating version because skaters can build up much more momentum before they jump. This isn't a criticism of dancers, it's just physics.