If I understand the Wiki article correctly (all this would be a lot easier to describe in pictures than words), the folding-in method is actually used in Japan (possibly among others). I certainly don't recognize it as American, but then I'm having a hard time just picturing any of this, so who knows.
The article also makes a distinction between counting something for yourself and counting so someone else can see it, which might be worth considering.
When I use my fingers to count absolutely for myself in my own head, as when counting syllables to check poetic meter (or haiku, or hymn meter), or seconds of time elapsed, or words in a sentence or whatever, I use my right hand* with the palm facing down and tap the fingertips down onto a surface (or on the palm of my left hand) in piano-fingering order (starting with the thumb), repeating while thinking or silently mouthing the actual numbers (or alternatively adding multiples of five in my head: 1 2 3 4 5, 2 2 3 4 5, 3 2 3 4 5, etc.).
*As a pronounced right-hander who hardly uses my left for anything, which may be partly why I never got very good at either piano or violin.
If I were counting partly to myself but also so that someone nearby could see, I think I might very occasionally (but not often) do it with what you call the folding-out method: starting by holding out my right hand at waist level in a loose fist with the palm facing up, parallel with the floor, and then opening in piano-fingering order (thumb first), with a little bounce down as each finger is added.
But to count visibly so that someone can see it from a distance, or to show numbers at a distance, as when gesturing across a crowded room to ask for a free table or a number of seats (or as a toddler showing how old one is), I would use what the Wiki article describes as the American method: hand held up at or above face level with the palm facing out, and holding up first the index finger (1), then index and middle (2, spread out, like the V for victory sign), then index middle ring (3), then index middle ring pinky (4), and last adding the thumb (5). For numbers above 5 I would use two hands, or flash one hand twice.
Not sure if any of that helps, but FWIW. (-:
I'm lagging many posts behind, having taken time out to write. But just glancing hastily at the quote, it doesn't seem typical to me for American counting. Is there by any chance any other reason why she might want to clutch in rather than opening out, draw the counted items to herself, so to speak; or why the writer might have wanted to emphasize that she was actually doing it in an unusual manner?