Since installing iOS 10 Public Beta 1 on my iPhone 6 Plus I was couldn't quite figure out how the new unlock/open scheme was supposed to work. I've been reading this thread looking for answers but it just seems like a lot of misunderstanding and confusion. Here is what I've figured out on my phone, hopefully others can confirm. My phone does not support raise to wake so I cannot test that, and it's TouchID V1.
In iOS 9, you could either press the home or power buttons to wake the screen, then rest your finger on the home button to unlock and open the home screen without a click. This behavior has changed and so far the new implementation seems really buggy and that's probably what's causing the confusion.
In iOS 10:
You can wake the screen by clicking either the power or home button, remove finger and the phone remains locked. Resting a finger on the home button unlocks but does not open the phone. Once it is unlocked, another click on the home button opens the phone to the home screen.
Sometimes, I have found that you can wake the screen by clicking the home button, remove finger, then rest finger on the sensor to unlock AND open the phone to the home screen without a second click. I have yet to figure out what conditions allow this. I would guess that this behavior is intended for when "Rest to Unlock" is enabled, or after "Raise to Wake?" I have never seen this work after waking with the power button.
OR
When the phone is asleep you can also click and keep your finger resting on the home button, which unlocks AND opens the phone. This seems to work reliably most of the time.
There is no way to unlock and open the phone without clicking the home button at least once, even after waking the screen with the power button. I see no difference whether "Rest to Unlock" is enabled or disabled even after multiple toggles and reboots.
Obviously these changes are intended to allow more interaction with the new notifications from the lock screen without opening the phone, but so far it's just really buggy. I'm sure as the beta testing continues the unlock/opening behaviors will be considerably refined. It's funny how such a small change can cause so much trouble, but not surprising since unlock/open is something we've all learned to do a certain way, habitually, several times a day for years now.