They discuss this in the paper. Scientists aren't just randomly doing shit, they do actually put in thought.

Video Playback Techniques

Pigeons are social animals. They often communicate with each other by exhibiting distinct behaviors, such as courtship displays in front of potential mates. Accordingly, the occurrence of a courtship display indicates that a bird ‘recognizes’ the visual object as a potential mate [Shimizu, 1998; Partan et al., 2005; Patton et al., 2010]. In studies in our laboratory, as well as that of Barrie J. Frost and Nikolaus F. Troje [Frost et al., 1998], subject pigeons were exposed to live stimulus pigeons, videotaped pigeons, and photographs of pigeons to examine which visual features can trigger spontaneous courtship displays.

...

The significant results of video-playback stimuli further emphasize the potency of visual signals in courtship, considering the fact that the quality of a video-playback stimulus is much poorer compared to both real live stimulus birds and taxidermic models [D’Eath, 1998]. The video-playbacks were presented on a 2-dimensional monitor screen that was designed for the eyes of human observers. Video stimuli cannot provide depth information nor accurate color information for the avian visual system. However, the fact that video-playbacks can trigger courtship responses does not imply that the stimulus deficiencies are irrelevant for the expression of courtship responses.

Edit: Also as others have discussed, the Tumblr OP is misrepresenting the results of the study. Even in this study the pigeons showed clear preference patterns that indicate that they're understanding that it's a picture of a pigeon.