Shouldn't we decide whether the $x$ is usual or unusual by looking at its probability?

That's one way to look at it. However, values can have low probability without being weird. For instance, when you roll a normal $D(6)$ die, there is a low probability of rolling a one...and a two and a three and a four and a five and a six. Each outcome has the odds against it in the sense that it is more likely than not for something else to occur: $P(X=x) = 1/6$ while $P(X\ne x) = 5/6$.

What your instructor describes is more along the lines of determining if that $x$ is within the mainstream of the distribution. If values are clustered around the mean and your observation is also near the mean, that is pretty typical.

This breaks down under some circumstances, however, such as when there is multimodality. For instance, consider a fair coin landing on its edge. That is, at least in some sense, the average on a heads flip and a tails flip, yet I would consider such a flip to be quite the anomaly.

This highlights some of the difficulties in "outlier" or "anomaly" detection. It's easy to identify a point as adhering to a rule that calls it an outlier, anomaly, or unusual point. The ease of determining such a rule is a different story. If there is a breakdown in the simple scenario of flipping a coin, imagine the complexities in a multivariate time series or in a video.