Summary:
Picture 1 is a piston monostable pulse generator that works fine.
Picture 2 is the same piston monostable pulse generator that doesn't work.
The only difference between the two is how the redstone signal input to the pulse generator is created.
Picture 1's input signal is created by powering the redstone via a block that is powered by a lever. (note: this also works just fine if the input signal is created by powering the redstone directly, e.g. with a redstone torch or a redstone block or directly by a lever or basically anyway except the way described with regard to Picture 2).
Picture 2's input signal is created by powering the redstone via a redstone torch that is powered when the block it's attached to is un-powered by the lever. That is a complicated way of saying that when you un-power the block by switching off the lever, the redstone torch attached to it turns on, thereby turning on the redstone and creating an input (albeit an input that doesn't work in this case).
What I expected to happen was that the redstone signal input to the piston monostable pulse generator would behave the same way whether it was powered by a power component directly or by activating a torch by un-powering the block it's attached to.
To duplicate, you can replicate the redstone mechanisms from the pictures. They're only one wide and you can see everything (the pistons are sticky, but I imagine you could use regular pistons and gravity blocks, although I didn't try that).
For what it's worth, this mechanism functions the same way on iOS and XBox.
Thank you.
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Confirmed and duplicated in 1.2.14.2. In other words, when the lever is powered, you would receive a 1 pulse tick into the lamp. If you place a Redstone torch on the lever-ed block, and power the lever (therefore turning the torch off), and turn the lever off (therefore turning the Redstone torch to active), you would get the piston to rise with the block, but no Redstone tick will pass through the repeater????

Confirmed and duplicated in 1.2.14.2. In other words, when the lever is powered, you would receive a 1 pulse tick into the lamp. If you place a Redstone torch on the lever-ed block, and power the lever (therefore turning the torch off), and turn the lever off (therefore turning the Redstone torch to active), you would get the piston to rise with the block, but no Redstone tick will pass through the repeater????

Confirmed and duplicated in 1.7. No Redstone tick is passing through the repeater if the Sticky piston is powered by a Redstone Torch and a lever. It only passes through the repeater if the piston is powered by a lever and Redstone dust exclusively.

Confirmed and duplicated in 1.7. No Redstone tick is passing through the repeater if the Sticky piston is powered by a Redstone Torch and a lever. It only passes through the repeater if the piston is powered by a lever and Redstone dust exclusively.

This is a duplicate of MCPE-15793 . Depending on what powers your device it may or may not work. Due to MCPE-15793, a world update (like flicking a lever directly connected to your circuit) will cause your machine to work as the repeater receives the power instantly. However when the device receives a signal from a redstone update, the repeater does receive power however this is instantly lost by the world update caused by the moving piston.
Quick recap:
World Update: RS Tick 1: power is received, instantly turns on repeater (instead of waiting 1 tick), resulting redstone updates occur. RS Tick 2: power is lost due to piston removing block, repeater instantly shuts off,
Redstone Update: RS Tick 1: power is received, Repeater Delay will wait 1 tick to turn on. RS Tick 2: Repeater activates, Piston removes block causing power to be lost instantly, ignoring the repeaters delay, shutting it off, no resulting redstone updates occur (as nothing really changed)

This is a duplicate of MCPE-15793 . Depending on what powers your device it may or may not work. Due to MCPE-15793, a world update (like flicking a lever directly connected to your circuit) will cause your machine to work as the repeater receives the power instantly. However when the device receives a signal from a redstone update, the repeater does receive power however this is instantly lost by the world update caused by the moving piston.
Quick recap:
World Update: RS Tick 1: power is received, instantly turns on repeater (instead of waiting 1 tick), resulting redstone updates occur. RS Tick 2: power is lost due to piston removing block, repeater instantly shuts off,
Redstone Update: RS Tick 1: power is received, Repeater Delay will wait 1 tick to turn on. RS Tick 2: Repeater activates, Piston removes block causing power to be lost instantly, ignoring the repeaters delay, shutting it off, no resulting redstone updates occur (as nothing really changed)