I live in the heart of the Amish/Mennonite world: Lancaster County.
There are Mennonites who are very contemporary. You wouldn’t really be able to tell them apart from anyone else. There are Old Order Mennonites who are as Old Order as the Amish can be. You can tell them apart by the buggies they drive. Amish drive gray, Mennonites drive black. There are Amish up around Lewistown, PA who drive white buggies. And I’ve heard of a group that drives yellow buggies.
The Mennonite men around New Holland, PA wear black straw hats that look like fedoras, and they ride bicycles.
Also, the women wear different bonnets. Amish bonnets are heart-shaped, Mennonites’ are round.
I’ve never heard the term Amish Mennonite, but I do know that all Amish are Mennonite but not all Mennonite are Amish.
And I think the Brethren are somehow connected.
Anyone ever see that ridiculous show, “Breaking Amish?” It was hilarious. Loved the part where the Amish guy didn’t know what a microwave oven was.

(of course he knew what it was.

)
Trivia–the Stoltzfus family was among the first (if not the first) Amish families to settle in Lancaster County. One Stoltzfus family. There are now an estimated 1 million Stoltzfuses in the U.S. descended from that one family (including Smuckers, Schmuckers, and other variations).