As a CTO myself, i see there is a lot of vagueness in the whole role of CTO. Almost all other roles in an organization are well-defined and have clear goals and responsibilities. The role of the CTO varies so widely between each organization, it is something that has to be specified clearly when going through the hiring process or even when starting a startup.
Why does this happen? I have found it has a lot to do with the investors and management team. I've seen the definition of the CTO varies as far as "The CTO should be the technical evangelist, super SEO." all the way to, "The CTO is in charge of Product and delivery."
IMHO giving CTO titles to the "evangelist, super SEO" undermines the role, because that role does not warrant either "Chief" or "Officer" in their title.
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As far as myself, i retain the CTO title because am the owner of the technology; in that i make sure that technology is all the things we are trying to achieve as a business; looking beyond the current implementation and plan out how we need to build the product over the coming years. This goes alway down to components required, system architecture, external interfaces and how we can do things better, faster, cheaper. I also write pieces of the code and help out in engineering task as needed. This keeps my competency in day to day ENG high. Involvement with ENG tasks and coding is what keeps my sanity.
If someone asked me if i liked my role as CTO i would say yes but there is all the things you listed (fight fires and battling with managers and tech team).