How to get the square wave output at Photodiode
There are several areas in which this can be complicated:

First of all the camera with have an integrating period, this will tend to blur out your wave form edges so your the precise timing won't be easily determined.  i.e. your edges will be blurred.  This is a consideration for when you get the circuit going.

You don't say what your display is, but it is entirely possible that your display is generating too much light in it's "off" state.  Most LCDs have a finite contrast ratio and leak light in the off state.  You can see this is a darkened room easily.  So if it always on then your photodiode is too sensitive or there is stray light interfering with the PD.

Maybe shroud the detector so room light cannot get in.

If you are getting only small amount of signal at your PD then it's possible that the display does not generate light for the photodiode.  You can solve that with additional circuitry (a gain stage) or with a phototransistor.