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When the 610 was first launched, there was an issue with the charging the unit, as detailed below:
I believe this was apparently solved by a firmware update. I'm having the exact same problems now, watch is only a few months old. Initially it shut down. I followed the master reset instructions (took a few attempts as there was barely any juice). After a lot of faffing around, I went to bed with 60% battery last night, up at 05:00 for my run and lo and behold, the battery went flat when "locating satellites". Plugged it in when I got home, and it displayed 45% battery charge, but is now reverse charging.
Laptop or wall mount, it is reverse charging. Anybody else recently had the issue? The forum search above dates back a couple of years...