I would think of it as the whole embedding concept again on a finer grained scale: you wouldn't say the model 'has a dimension of whether the input is king', instead the embedding expresses the idea of 'king' with fewer dimensions than would be needed to cover all ideas/words/tokens like that.

So the distinction between a direction and a dimension expressing 'gender' is that maybe gender isn't 'important' (or I guess high-information-density) enough to be an entire dimension, but rather is expressed by a linear combination of two (or more) yet more abstract dimensions.