What is it that you think a CTO does? There isn't a standard answer to this question.

CTO is usually the exec responsible for the entire tech org. The CTO reports to the CEO, and the top managers and maybe a few ICs in the tech org report to the CTO.

I was CTO of a <20 person startup. I recruited the entire tech team, collaborated with the CEO to build the product backlog and spec things out, presented to investors, but also had at least 50% time to code. Not all “CTO” roles are the same. At a small company they better be hands on.

This is very similar to my own role, but I didn't have (nor would I have accepted) that title.

On my resume, I usually list it as “Lead Engineer” since it fits the roles I’m applying to better.

I think a founder/early gets away with "CTO" on their resume, esp. if they're the only person in the org with the role (ie: it's a PM-style CTO, and there isn't a VP/PM; or: it's a VP/E-style CTO, and there isn't a VP/E). But outside that circumstance, given the choice, I'd rather have the VP/PM or VP/E role than "CTO".

(As we get deeper into these threads I am further out on a limb.)

Yes, I was one of the first hires. My role was closer to VP/E and "CTO" was mostly vanity, a reward for being early and getting a new company through the first couple years.

It's very common to see a VP of Engineering managing the day-to-day operations while the CTO acts in a capacity like this.

I’ve seen that too, but then the VP of Engineering tends to report to the CTO, and not to- say, the CEO directly.

Dotted line reporting is very different. In these instances the VP/E is usually directly interfacing with other executives as the CTO's peer. This is even more true when the budget is managed by the VP/E and the CTO is more customer/sales facing.

You're saying "usually" about something that has definitely not been a norm in my career. It seems like there's really only two ways to interpret that arrangement: either the CTO is in fact the EVP/E (fair enough! lots of CTOs are other exec roles with a funny hat), or the CTO has a single top-level manager report, in which case what really happened is that the org hired a pro to run engineering and put the "CTO" out to pasture.