I've been working on a paper on nonresponse bias. As part of the setup, we describe the MAR and NMAR mechanisms first defined by Little and Rubin. In short, Missing-at-Random means we can fix the bias due to the missingness using the available data. While Not-Missing-at-Random means we can't repair the bias with the available data.
It can be hard to discuss the problem with this divide. We were looking at situations where the bias could be smaller and it could be bigger. The NMAR/MAR distinction doesn't capture that very well. There is another formulation that is actually pretty good for discussing different degrees of bias remaining after adjustment. It's due to the following article (I couldn't find it online):
Kalton, G. and D. Kasprzyk (1986). "Treatment of missing survey data." Survey Methodology 12: 1-16.
They define bias as having two components: A and B. One of the components is susceptible to adjustment and the other is not. In some situations, you can reduce the bias due to A and there will still be the bias due to B. In other situations, by adjusting for A you can actually increase the bias (i.e. when A and B are equal and in opposite directions).
In any event, I think this is useful for talking about bias as it gives us a way to talk about more or less biased that the NMAR/MAR distinction does not.