Amish come from Mennonites. In the 16th century when the Amish split from the Mennonites they began to enforce stricter and more conservative policies (rejection of technology at this point includes untrimmed beards, hooks vs buttons, certain farming tools, plain dress etc), which is the basis of the Amish today. When electricity was involved that too was shunned but the split actually occured when Jacob Ammann formed the first Amish community. There are also old order Mennonites who live similarly but not as strictly. So I would say they began to be distinguished as separate at that point in time, because the whole creation of the Amish was a distinct rejection of technology and intentional distancing from the community.
I don't post here ever and hope this is okay without a source - this is all pretty standard Mennonite history I have used for this conclusion.
E for clarity
They don't reject technology, and they don't think it's evil; they just selectively adapt it. There is an unwritten set of rules called the Ordnung that dictates this, and it's administered and maintained by each church. There's no one answer to this; you'd have to look at it in a case-by-case and church-by-church basis.
There was a good article in IEEE's magazine about ten years ago about this. Here's a copy: http://archive.cspo.org/documents/Wetmore-AmishTechnology-v2.pdf
The John Hostetler book Amish Society mentioned below is also good, as is Donald B. Kraybill's The Riddle of Amish Culture
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