It's the output of the activation function.

I think the counter-examples that you cite are just omitting the function for brevity, or because it doesn't matter in context, or people are just inconsistent and that's fine.

Personally, I would say that the neuron's activation is the output of the activation function. But there is clearly inconsistency in the literature, as you show. So how do I justify my intuition against highly-important published texts such as Goodfellow et al., 2016.

To make the case for why it should be the output, we can first draw from the nomenclature in neuroscience, as this is where we derive most of the terms from in the first place. For example, A. Hoffmann (2001), writes (quote found here):

Neural activations are mostly stimulated circularly. A neuron is activated by other neurons to which it is connected. In turn, its own activation stimulates other connected neurons to activation.

Here, clearly, the activation of the neuron is what stimulates antecedent neurons, i.e. it must be the output of the neuron.

Secondly, despite the naming of the mathematical notation, Goodfellow et al. also allude to the same conclusion in this passage (pp. 165):

Each unit resembles a neuron in the sense that it receives input from many other units and computes its own activation value.

Finally, in the passage that you cite from Goodfellow et al. (2016), they do make reference to $a^{(k)}$ as being the "pre-nonlinearity activation" (despite referring to it just as the "activation" in the algorithm description). The fact that this is being explicitly specified implies to me that the other "kind of activation" is the primary use of the term.