Ok I think I have an answer. This happened to me twice and both times, it resolved the same way.
CAUSE: Both times, this happened when I let the battery run down too much. I was playing a game or listening to music and ignored the low battery warnings thinking I could push it and eventually the iphone shut down. No problem, I thought. I can just plug it in and keep on going. But when I plug it in, the iphone won't charge ... like it normally does. I get the red battery meter and the plug and lightning symbol as previously described. The phone won't turn on. I tried resetting it several time - hard resets as well - with no response. Always to the screen with the red battery.
SOLUTION: What i did both time was to keep the phone plugged in for at least 4 hours and maybe even more. I noticed that the red battery symbol flashes when it is not plugged in but when it is plugged in - the red battery symbol does not flash. Even though the phone is non-responsive in every other way, it seems to recognize if it is plugged into a power source or not. So I kept it plugged in for several hours. By the way, the phone is totally unusable during this time.
After that time period - I did a hard reset and the phone booted up with the white apple symbol and I started recharging again - this time it took.
I did this TWICE so I think it will work in the future.
THEORY: I am not a computer person (I'm a gynecologist in real life) but I will try to explain this behavior as best as possible. I think when the battery runs down too low it shuts down. But it shuts down EVERYTHING! Even the software that charges up the battery - which is a bug that Apple needs to fix - but everything shuts down. So this may sound stupid but it seems that when it is shut down to this degree the phone does not know how to charge itself. That is why nothing works. The phone remains shut down. It does not reboot. It does not charge up. However, as noted above, the phone does seem to recognize when it is plugged in to a power source or not as indicated by the flashes on the red battery symbol. It flashes if it is not plugged in and it stops flashing when it is plugged.
Anyways, even though the phone does not know how to charge up, it is charging anyways - but only by a trickle. A microscopic trickle. And it takes a couple of hours via this trickle to charge up the phone to a certain level to be able to reboot. After the reboot - you have to charge up the battery again - but since the phone is now working, it knows how to charge itself up normally.
Again, this is an amateur theory but I can't explain this behavior any other way.