The model you referenced in your question is called the "one-way model." It assumes that random row effects are the only systematic source of variance. In the case of inter-rater reliability, rows correspond to objects of measurement (e.g., subjects).
One-way model: $$ x_{ij} = \mu + r_i + w_{ij} $$ where $\mu$ is the mean for all objects, $r_i$ is the row effect, and $w_{ij}$ is the residual effect.
However, there are also "two-way models." These assume that there is variance associated with random row effects as well as random or fixed column effects. In the case of inter-rater reliability, columns correspond to sources of measurement (e.g., raters).
Two-way models: $$ x_{ij} = \mu + r_i + c_j + rc_{ij} + e_{ij} $$$$ x_{ij} = \mu + r_i + c_j + e_{ij} $$ where $\mu$ is the mean for all objects, $r_i$ is the row effect, $c_j$ is the column effect, $rc_{ij}$ is the interaction effect, and $e_{ij}$ is the residual effect. The difference between these two models is the inclusion or exclusion of the interaction effect.
Given a two-way model, you can calculate one of four ICC coefficients: the single score consistency ICC(C,1), the average score consistency ICC(C,k), the single score agreement ICC(A,1), or the average score agreement ICC(A,k). Single score ICCs apply to single measurements $x_{ij}$ (e.g., individual raters), whereas average score ICCs apply to average measurements $\bar{x}_i$ (e.g., the mean of all raters). Consistency ICCs exclude the column variance from the denominator variance (e.g., allowing raters to vary around their own means), whereas agreement ICCs include the column variance in the denominator variance (e.g., requiring raters to vary around the same mean).
Here are the definitions if you assume a random column effect:
Two-way Random-Effects ICC Definitions (with or without interaction effect):
$$
ICC(C,1) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_{rc}^2 +
\sigma_e^2)}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + \sigma_e^2}
$$
$$
ICC(C,k) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_{rc}^2 + \sigma_e^2)/k}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + \sigma_e^2/k}
$$
$$
ICC(A,1) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_{rc}^2 + \sigma_e^2)}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_e^2)}
$$
$$
ICC(A,k) = \frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_{rc}^2 + \sigma_e^2)/k}\text{ or }\frac{\sigma_r^2}{\sigma_r^2 + (\sigma_c^2 + \sigma_e^2)/k}
$$
You can also estimate these values using mean squares from ANOVA:
Two-way ICC Estimations: $$ ICC(C,1) = \frac{MS_R - MS_E}{MS_R + (k-1)MS_E} $$ $$ ICC(C,k) =
\frac{MS_R-MS_E}{MS_R} $$ $$ ICC(A,1) = \frac{MS_R-MS_E}{MS_R +
(k-1)MS_E + k/n(MS_C-MS_E)} $$ $$ ICC(A,k) = \frac{MS_R-MS_E}{MS_R +
(MS_C-MS_E)/n} $$
You can calculate these coefficients in R using the irr package:
icc(ratings, model = c("oneway", "twoway"),
type = c("consistency", "agreement"),
unit = c("single", "average"), r0 = 0, conf.level = 0.95)
References
McGraw, K. O., & Wong, S. P. (1996). Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1(1), 30–46.
Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: Uses in assessing rater reliability. Psychological Bulletin, 86(2), 420–428.
