$R^2$ for mixed models = ICC?

I will be referring here to Nakagawa and Schielzeth (2013). As those authors state, $R^2$ for OLS regression could be defined as follows:

$$R^2 = \frac{\sum^n_{i=1}(\bar{y} - \hat{y_i})^2}{\sum^n_{i=1}(y_i - \bar{y})^2} = \frac{var(\hat{y_i})}{var(\hat{y_i})}$$

and so, if we define mixed model like this:

$$y_{ij} = \beta_0 + \sum^p_{h=1}\beta_hx_{hij} + \alpha_{j} + \epsilon_{ij}$$

then $R^2$ could be defined as follows:

$R^2$ for fixed effects ($R^2_m$):

$$R^2_m = \frac{\sigma^2_f}{\sigma^2_f + \sigma^2_\alpha + \sigma^2_\epsilon}$$

where:

$$\sigma^2_f = var\left( \sum^p_{h=1}\beta_hx_{hij}\right)$$

$R^2$ for random effects ($R^2_c$):

$$R^2_m = \frac{\sigma^2_f + \sigma^2_\alpha}{\sigma^2_f + \sigma^2_\alpha + \sigma^2_\epsilon}$$

However, this seems to be the same as ICC:

$$ICC = \frac{\sigma^2_\alpha}{\sigma^2_\alpha + \sigma^2_\epsilon}$$

and so:

$$R^2_c - R^2_m \approx ICC$$

This puzzled me a bit. Could you correct me if I am wrong? Why ICC is the same as $R^2$ for mixed models? If those two measures are equivalent, why are they not used like this? (or if they are, could you provide a reference?) Could you explain why they are equivalent?

Reference: Nakagawa, S. & Schielzeth, H. (2013). A general and simple method for obtaining $R^2$ from generalized linear mixed-effects models. Methods in Ecology and Evolution, 4, 133–142.

Nakagawa and Schielzeth (2013) call $R^2$ as "closely connected to $R^2$" and refer to their 2010 paper. From posting this question I already seen few mentions of $ICC$ treated as "variance explained" since it in fact is a part of total variance explained by all the random effects and residuals. So it seems $ICC$ is used as an analog to $R^2$ while one have to remember that both statistics have their flaws (e.g. here and here).