I've only used ARL and like them very much so I can't compare their lab with TEI. One thing I find super helpful is that Dr. Eck determined what the ideal value of each mineral should be and the ideal ratio. I think the ARL site, www.arltma.com, explains his method. In my quest for wellness, it was always frustrating to go to physicians who did tests, usually blood and said,"you are in the normal range" while I felt lousy. I love having hair analysis show the ideal and the fact that my program and practitioner are aiming for that.
Regarding copper, I was high from my very first test, but my husband wasn't. There are indicators that show that he has a hidden copper toxicity, and, sure enough, as time goes back, he's been dumping copper, too.
BTW, there is also a FB group where we all love ARL.