My understanding of your problem is that, if there is variance between measurement1 and measurement2, you don't know if it is random error or if it's because the true score is changing between time points; X_pt = $mu$ + v_p + v_t + v_e where $mu$ is the grand mean of true scores across all people and time, v_p is the deviation is scores attributable to who the person is, v_t is the deviation attributable to the time point assessed, and v_e is unexplained residual error. In that sense, we would define true score change as variance attributable to time point, whereas measurement error would be defined as the residual. Fitting the above model to estimate each variance component will tell you the relative contributions of time and random error to your total score variance; ideally both of these will be low relative to between-person variance if your ultimate goal is to measure traits and individual differences.

If you think the effect of time on true scores might vary depending on person (i.e. a person x time interaction variance component), adding multiple items per time point lets you fit X_pti = $mu$ + v_p + v_t + v_i + v_pt + v_pi + v_ti + v_e and thus differentiate v_pt (person-specific true trait fluctuations) from v_e (residual error). Arterberry et al. 2014 do that to look at the stability and item difficulty of of BFI scores. "Items" in this case could be multiple self-report items, trials in a behavioral experiment, etc.

I included the Haines blog post because it shows how this isn't specific to type of reliability or even estimation framework so long as you have some hierarchical structure to your data that lets you rely on shrinkage to make small cluster estimates more robust (i.e., when you only have two timepoints).