The Bellamy Salute was the official way to salute the US flag under Flag Code, 1892-1942 ... although I forget why we changed it in 1942?
Photo is from 1915.
1942 photo of US schoolchildren in Southington, CT offering the Bellamy Salute. This was the last year it was official Flag Code.
Fascists adopted the open handed gesture because it was widely believed to be a victorious gesture used by Ancient Romans (it wasn't).
*This* is in fact the Roman Salute (sources vary!), with three fingers touching the palm, although a perfunctory salute might not have had the full motion, and might indeed have looked like just an open palm.
I'm sure there are plenty of references to fist or palm to chest, then out. The US military has what's called the "guidon salute", which is a non-formation flag-bearer's salute.
It's the kind of thing that is poorly written about, because it was assumed everyone was familiar with it.
There's no real consensus on what the Roman salute looked like. There are statues of it, sure, but there is no description of it from antiquity so we have no real information other than they had a salute. That said, I agree the Nazi salute is not the same thing.
