I think the confusion is semantics. Most people think of a startup as a growth business. A startup is different from a small business. Whereas a small business may aim for $10M in revenue per year in 5 years (and maintain that), a startup may aim for $1B in revenue per year in 5 years.
So, if you're the CTO in your second year of business and still writing most of the code, you're probably a small business.
That said, in my experience, if you're a CTO and not able to at least understand the source code, you're a liability. I've worked for CTOs like that, and they were unable to manage risk or estimate beyond random guesses. And, it's unlikely that such a person can forecast or expand business safely without at least some trusted advisors.
I think Paul Graham uses the term "scalable". But I have seen talking points from lots of big VCs and other figures in business who seem to think it's more about age, profitability, and management structure that define a startup.
To focus on the scalability point, I tend to agree that many only think of scalable businesses as startups, but I think the predicted numbers don't have to be in the billions to be considered a scalable growth business.
If you are a one man shop building an innovative product and you raised seed money from friends and family and hire 2 employees and expect to start making $1m a year as a 3 man shop in a few years, I think most people would call that a startup.
On the other hand, if you raise funds and build just another restaurant on the corner but with no innovation in either product or scale, then I think you are a small business.
You made me curious, so I did some quick investigation on the etymology of the word. :) (sorry, insomnia)
Using Google NGram viewer, I looked up "startup" and quickly noticed that the word (in modern usage) seemed to be short for "startup costs". Interestingly, "startup costs" seemed to only come into use after the 60s and 70s and was often connected to financial instruments. So, I also plotted the word "options" and was surprised that the shape of that curve was the same as the curve for "startup". They just had different magnitudes.
I also learned that stock options became tradable instruments in the 70s. And, also, startups began being formed in the 70s. That is, they seemed to go hand in hand. In particular "startup" was used to describe speculative research and development ventures with high capital requirements.
More recently though, the word "startup" became being associated with small teams, minimum viable products, and micro seed funding. So, that could be what you mean.
So, if you're the CTO in your second year of business and still writing most of the code, you're probably a small business.
That said, in my experience, if you're a CTO and not able to at least understand the source code, you're a liability. I've worked for CTOs like that, and they were unable to manage risk or estimate beyond random guesses. And, it's unlikely that such a person can forecast or expand business safely without at least some trusted advisors.