Omitted variable bias is very specific and refers to the bias in the parameter $\beta$ when omitting the variables $Z$ from a regression model of the following form:
$$Y = \alpha + X\beta + Z\gamma + \varepsilon
$$
It is a purely statistical concept in that it has nothing to do with the causal relationships among the variables. As long as $X$ and $Z$ are correlated, there will be bias in estimating $\beta$ when $Z$ is omitted.
However, $\beta$ does not always represent a useful causal parameter. For example, if $Z$ is a mediator, then $\beta$ represents the direct effect of $X$ on $Y$. The direct effect is not always the quantity of interest. If the total effect is of interest, and $E[X\varepsilon]=0$, then omitting $Z$ from the model yields a biased estimate of $\beta$ but an unbiased estimate of a different parameter, the total effect. On the other hand, if $Z$ is a confounder, then $\beta$ represents the total effect of $X$ on $Y$, and omitting $Z$ would yield a biased estimate of the total effect. So, while omitted variable bias does not concern the causal status of the variables, the choice of whether to omit $Z$ (or the consequences of being forced to omit it) depends on the causal status of the variables and the quantity of interest.
More broadly, the model for $Y$ above doesn't even have to represent a valid causal model for omitted variable bias to occur. If it described the relationship between the variables in a general multivariate distribution (e.g., if $Y$ precedes $X$ and $Z$ but the relationship among them is accurately represented by this equation), then omitted variable bias can still occur if $X$ and $Z$ are correlated even if $\beta$ does not correspond to a meaningful causal parameter (which it wouldn't if $Y$ precedes $X$).
I think one thing missing in econometrics training is linking the statistical model to the causal parameter of interest; it is assumed a coefficient in the structural model for an outcome is a causal parameter of interest, when in reality the interpretation of that parameter depends on the causal relationships of the variables in the structural model, which are not captured by the structural model for the outcome. So, while omitted variable bias describes a real feature of regression, it doesn't directly map onto the more useful causal concepts of confounding and endogenous selection. In addition, these concepts are not attached to any specific model and don't require functional form assumptions to discuss, whereas omitted variable bias usually refers to the specific bias incurred when employing a specific statistical model (e.g., linear regression).
