I guess you have come up with an answer by now, but Tjur's $R^2$ applies only to binomial GLMs. To calculate Tjur's $R^2$, we estimate the average predicted probabilities for each category of the dependent variable, and then take the absolute value of the difference between them. So, for two categories the interpretation is rather straightforward. As Allison explains, "if a model makes good predictions, the cases with events should have high predicted values and the cases without events should have low predicted values." For ordinal or multinomial logistic regression models, the generalization of Tjur's $R^2$ is not straightforward. However, there are some attempts. For example, Smith et al. (2021) proposed an extension for multinomial and ordinal outcomes:
$$D' = \frac{\sum_{i=1}^K|\overline{\hat{\pi}}_{1i} - \overline{\hat{\pi}}_{0i}|}{K}$$
"where $\overline{\hat{\pi}}_{1i}$ and $\overline{\hat{\pi}}_{0i}$ refer to the mean predicted probability of being in category $i$ for cases corresponding to response category ‘$i$’ and ‘not $i$,’ respectively; and where $K$ denotes the total number of outcome categories." They argue that
Similar to Tjur’s coefficient of discrimination for binary
logistic regression, $D$′ takes on large values (maximally, unity) for
situations in which the nominal or ordinal regression function
maximally discriminates among the outcome categories, and takes on
small values (minimally, zero) for situations in which the model
provides minimal discrimination.
Smith, T. J., Walker, D. A., & McKenna, C. M. (2021). A coefficient of discrimination for use with nominal and ordinal regression models. Journal of Applied Statistics, 48(16), 3208-3219.
