Brianna,
Thanks for bringing the WASABE paper to my attention. Interesting work. They are reporting on an audit "instrument", ie a multi-domain tool to assess neighborhoods. For conceptualizing what is measured, note that each domain is really a separate construct, for example: "Transportation environment: Features that facilitate safe and efficient movement and active transportation throughout the environment including traffic volume, street type, presence of sidewalks and bike lanes, and presence of public transit." Is one construct.
They state that they assessed validity by evaluating whether the measure correlated with other measures as expected based on theoretical understanding: "We also found the instrument has good construct validity, as most significant differences in presence or absence of features were found in the direction that one might expect". This is called "construct validity" (as opposed to other approaches to assessing validity, such as face validity or criterion validity). It is among the most common and important ways to evaluate the validity of a measure, ie whether the instrument is measuring what you think it is measuring.
The Perry et al paper is not actually describing a disparity. This is also a paper about developing a measurement tool. The goal of the tool is to help people assess disparities, but this particular paper presents no reports on disparities that I can find in the text.
Maria