This is an interesting topic ... and definitely can cause some confusion.
It is also my understanding that the early American salute was similar to the present-day British salute -
https://www.quora.com/What-is-the-difference-in-the-traditions-of-the-outward-palm-and-the-palm-down-salute This is cropped from the above link -
Ironically, although one seldom sees it mentioned in discussions like the one linked to above, until well into the 19th Century the British Salute was actually a "palm down" version, much more like the present-day American salute, although with a somewhat bent wrist and fingers rather than the "dead-straight" American version -
The above illustration also demonstrates something else about British saluting which is not commonly known but remained unchanged until about WWI - namely that the salute was rendered with
either the left or right hand, depending on which side was closest to the object of the salute! (And, if not wearing headdress, the soldier simply came to attention and did not salute.)
Note this description of the above type of salute from the 1859 "Field Exercise & Evolutions of Infantry" ... although the written description does seem to describe a somewhat "neater and tighter" form of salute than is shown above. I also have access to an 1861 version of the "Field Exercise & Evolutions" in which the description is the same -
I am not certain exactly when the British salute changed to the present form, but it had happened by the time of the 1870 version of the "Field Exercise & Evolutions" -
I
suspect that this change was fairly new in 1870, because of the need to specify that the palm should face outward and the inclusion of an illustration of the salute. I will go on the British "Victorian Wars Forum" and see if i can find out the precise year of change ...
My point about British saluting with either hand (or not saluting at all if uncovered) brings to mind a story I heard about some Victorian-era British reenactors in attendance at an event in the United States who were approacehed by an American "officer" who stood directly to their front. The man on the right saluted with his left hand, the man on left saluted with his right hand and the man in the middle, who happened to be hatless, simply came to attention. When the "know-it-all" took it upon himself to dress them down for such sloppiy and inconsistent saluting, they took great delight in informing him in no uncertain terms that they had all acted in strict and correct accordance with British military regulations ...