A lthough teaching certainly falls under the category of speaking, I prefer to think of it as consulting, that is, shaping topics to the interests and needs of an audience, preparing comments that will open new possible ways of approaching old challenges, and prompting the listeners to engage in a conversation with feedback, comments and questions.
Indeed, it could be its own category altogether as there is something particularly enjoyable about teaching that is not found in public speaking, consulting, or writing alone. It is, in a sense, a combination of all three, while also allowing for an immediate interaction that is both challenging and exhilarating.
I suppose that should come as no surprise. Teaching these subjects at higher levels is little more than providing in-depth consulting to an audience motivated enough to show up voluntarily.
(Recent lectures and classes I have given, or will soon be giving, are listed here.)