Midnight-shadow
3rd Black Belt
yes and no.
sanchin (by any name) is a very popular form from the Fujian provence of China. there are multiple versions of it all different but all the same at the same time. white crane: Bai He Quan is only one style that uses this form. there are many many others. it is well documented that the Okinawan form sanchin: Goju- ryu , Uechi -ryu and Ryuei- ryu all share this form (as well as another from called Seisan) and is known to be passed down from Fujian styles. which styles? we dont actually know. there were many lineages of styles that did not have names. Bai He Quan was A style that was well known. often if someone was asked what style did they do, rather then have to explain it was a no name style they would rattle off the name of a well known style. there was a Tea merchant named Gokenki who was friends with Chogen Miyagi (creator of todays Goju) and Kanbun Uechi (founder of uechi ryu) who was a white crane stylist. some speculate the crane features of these styles come from him. my guess is that this is not the case. All three founders of these styles traveled to China and studied there and this is where sanchin by any name comes from.
beyond these three men and their respective styles is where the white crane influence ends. these three styles are loosely knows as Naha Te. but there was also Shuri Te and Tomari Te styles of karate neither of which have any relationship to any known Chinese styles, especially white crane. as a note modern Shotokan is based on Itosu's Shuri Te. most styles of Okinawan karate do not have a link to white crane kung fu. the Shuri and Tomari Te styles are much older than the Naha Te styles. the Shuri styles while probably did originate from China had a long time to develop and were used by the Pechin aristocratic class who would have regular official business with China. There were also Chinese who settled in Okinawa and it is thought some arts where brought in that way as well.
So, do you think it is more likely that Sanchin was just a single form that was taken from Fujian and incorporated into already established Karate styles? I guess it does make sense now that I think about it more.