CTO: I talk to my CEO and work out how technology can improve our business. Then I design what we're going to build, build it, and set it up to run on our servers.
If you're a CTO, do you actually do the things you claim in your second sentence, or do you tell others to do those things? I would expect the latter unless you're in a very small company, in which case "CTO" is probably an exaggerated title. (Eg: it really means "the tech guy" but you're incorporated so CTO is technically accurate.)
One of my pet peeves has always been managers describing what they do in terms of the work done by their reports rather than what they actually do themselves. It's not like managing isn't a lot of work, at least not for good managers. Why do managers take credit for others work instead of saying that they manage a team who accomplished X, Y, and Z?