Strange Waveform at high frequency from transimpedence amplifier
I was able to measure noise from the output waveform by applying a
  small signal to the non-inverting terminal, while the inverting
  terminal receives dark current.


The photodiode has capacitance. That capacitance forms a signal potential divider with the 1 Mohm feedback resistor so, at progressively higher frequencies, the non-inverting gain gets higher. Eventually the output clips and you get a distorted waveform.

At low frequencies the non-inverting gain is close to unity but, if your photodiode has 10 pF self-capacitance then the gain starts rising at about: -

Fc = \$\dfrac{1}{2\pi R_F C_D}\$ = 15.9 kHz



Generally, this problem is refered to as "noise gain" in TIAs.

See this answer about noise gain in photodiode amplifiers