We have a winner. I've been using lithium batteries on and off for the last few years on these. Lithium batteries were sitting in BOTH of these headlamps for the last six months. The turn on/shut off happened with those lithium batteries, and then with brand new alkaline batteries afterwards.
My guess is that the damage occurred in use with lithium batteries, not in storage. New lithiums can put out 1.8V which is too much for some devices. (In contrast alkalines put out 1.55V and NiMH/NiCad 1.35-1.40V when fresh.) The voltage drop under load (the technical term is "internal resistance") is also less than that for alkalines and NiMHs. (NiCads also have a very low internal resistance but, of course, start with a lower maximum voltage.)
See
http://data.energizer.com/PDFs/l91.pdf for some discharge curves. (Top of pg 2.)
I'd have to make measurements (current, etc) on undamaged and damaged headlamps to narrow down the nature of the damage.
Should lithium batteries not be used with any LED devices??? I just read a few articles and it suggested that lithium batteries shorten the life of headlamp LED bulbs. Maybe it's best to avoid using lithium batteries completely with headlamps?
This problem is individual device model dependent. (Petzel issued a warning for a number of its headlamps.
http://www.petzl.com/en/outdoor/news/products-news-0/2009/03/17/lithium-battery-information REI repeated the warning on its web pages.) Presumably it is a risk for any electronic device designed before (or ignoring the use of) lithium batteries. I've seen it/heard of it for certain cameras, headlamps, and GPSes. Many of the more recently designed devices are fine with them.
The (REI web page) description of the Myo RXP specifically states that it can be used with lithium batteries.
There is a workaround--run the lithium battery down a bit (in a tolerant device) before using it in a sensitive device. However, this reduces the useable capacity of the battery in the sensitive device. (I'd use a voltmeter before and after the pre-discharge verify that the initial voltage peak had been removed.)
For devices produced since lithiums became popular, you have to read the spec sheets or fine print. A web search may also bring up reports of damage (particularly for devices designed before lithums became popular).
Lithium batteries can also shorten the lifetime of incandescent bulbs.
Doug