I agree with a lot of what you wrote, but I do disagree quite strongly on your first (main?) point, as I understood it.
You wrote: "[...] a start-up is more of a determination after the fact than something you decide to build."
I totally agree with your point about companies normally not being able to "grow fast", and I agree with you that luck is a major factor. However, I think you're wrong in saying that people don't try to deliberately build "grow-big" companies.
Believe me, I'm a proponent of building small-growth "bootstrapped" companies, and I've had many conversations with founders about going that route. The vast majority, who had it in their heads that they wanted to be a startup, were deliberate in pursuing "grow-big" strategies. As one easy example, almost every company that raises VC money is either implicitly or, often, explicitly, chasing a grow-big strategy.
I completely agree that the vast majority of companies end up not being able to grow big, and have to decide between becoming a small company and "milking the cow" as you put it, or closing the company. This doesn't mean that they didn't set out, in the first place, to build a "grow big" company! In fact, you can see that this is true in the way that so many of these companies choose not to continue the small business, which could be profitable, but instead choose to fold the company instead.
In fact, reading the writings of YC, or talking with most honest VCs, will easily prove my point - they constantly tell you they are aiming to fund companies striving for $1b+ valuations, and they're fine with 99% of the startups they fund not getting there as long as 1% succeed. They are quite explicitly optimizing for "grow big" strategies.
Btw, I'd like to point out nl's comment below about Steve Blank's definition of a startup, which I always thought was the absolute best definition. A startup is an organization in search of a business model. When someone opens a pizzeria or a software consultancy, say, they're not searching for a business model - they're just executing one. But VC-backed "grow big" startups are very explicitly trying something new.
EDIT: Just to make sure we're not arguing over definitions - I'm claiming two things:
1. That what people, especially on HN, mean when they startup is a company pursuing a "grow big" strategy, and deliberately so vs. a non "grow big" strategy.
2. Regardless of 1, I also think that the majority of modern-day companies (let's say founded in the last 20 years?) that have grown big were deliberately started with the intention of growing big, ala Amazon, as opposed to say FB or Google, which wouldn't fall into this category. I'm less sure about this point, but it is a factual question that we should probably be able to answer.