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I do not know what these Amish refer themselves as but there is a group referred to as New Order.

What’s the difference between New Order and Old Order Amish?

Thanks for the links.
I don't think I've ever met a "New Order" Amish and your first link says they are limited to fewer than 20 church districts.
So we're talking about maybe 2,000 total out of nearly 300,000 overall = less than 1%.

Your first link also confirms my previous statement that New Order Amish retain far fewer of their young people (2/3 or less) than Old Order settlements (80% to 90% or more, and much more than 90% is my impression).
This implies a higher failure rate of settlements themselves which confirms my point that "New Order" often either leave the Amish or return to their Old Order roots.

Your Wikipedia article says 4,000 New Order in 65 districts, but also lowers their retention numbers to 1/2 or 2/3.
Also Wiki says that 35 of those 65 districts are "non-electric" which means that for nearly all practical purposes "New Order" is just a name for them and in fact they live quite similar to other Amish.
Regardless, they are still few enough that I'm not surprised at never knowingly meeting one.

Bottom line: "New Order" is more often a half-way house on the path to becoming "English", and that's really why there aren't very many of them.