All of this didn't work for me.
First of all : The trick about simultaneously pressing Home and Power buttons will potentially ''restart'' your iPhone 6, not ''reset'' it as many posters here says. I'm not a native English speaker but for me there's an immense difference between those two words. The word "reset" to me implies that something is lost (settings), while the word "restart" doesn't mean that. Bottom line: Don't be afraid of this procedure. You won't loose anything.
Anyway: here's what worked for me: The thing is that is is not at '''first''' black screen where you need to release the two buttons immediately, it is at the '''second''' black screen. Capice?
Do like this:
# Press Home and Power buttons simultaneously and hold them
# Screen will go black after some seconds. (continue to hold the buttons)
# Screen will show Apple logo after some more seconds (continue to hold the buttons)
# Screen will go black after some more seconds. Now you must immediately release those buttons.
# The phone is now turned off.
All together you need to wait a considerable amount of time. Perhaps something like 30 seconds.