I've been at several companies that have a CTO and a Director of Engineering. The CTO sets the strategy, and the Director of Engineering handles the execution. In theory the Director "reports" to the CTO (I.e. is under in the org chart), but not necessarily. Sometimes the Director reports to the CEO, and/or takes a more collaborative role with the CTO.
This does not apply at my current company, where the CTO has their title as an artifact of how the founding team was structured, but if I was founder/early at a company, progressed to a senior role, and then was told that I should take a role where I "set engineering strategy", I would immediately conclude that I was being managed out. "Strategy", in particular, is the kiss of death.
Both in my own personal direct experience and in 15 years of consulting, primarily for tech startups, the modal CTO I encountered had in reality a product manager role with a special title that was helpful in important pre-sales meetings --- and they did not tend to be the de facto VP/PM.
I think CTO as "other, better-defined kind of exec, but with a funny hat" is a perfectly cromulent model. I think CTO as "PM that customers feel flattered to talk to" is another perfectly cromulent model. To me, "CTO" is almost an honorific, or like a title of nobility.
