I like your short version except for "hire a contractor to build it." What if the company actually hires on one or two engineers to build it? Would that company have made the jump from "non-tech" to "tech"?
I wonder if it's simply a matter of engineer/development headcount i.e. what job function has the largest headcount. Google's is R&D (https://www.quora.com/How-many-employees-does-Google-have) and Microsoft's is Engineering (https://news.microsoft.com/facts-about-microsoft/#RevenueHea...). I tried to quickly find examples of some non-tech companies' (Walmart, Target, BofA, Wells Fargo, Nike) headcount by function, but couldn't.
Typically companies that have that technology component are able to respond faster, more creatively, and in a much harder to compete with way than companies that buy their base technology from third parties. The get a valuation boost from that.
That may be one way to look help define it.
However, in Redfin's case, it's simpler than that. The 3 founders[1] are David Eraker (software programmer), Michael Dougherty (electrical engineering & programming maps), and David Selinger (ex Amazon data mining).
Since the founders are still active and dictate the company's "DNA"... it looks like a tech company to me.
Similar to Jeff Bezos (computer science with some programming before D.E. Shaw) starting a tech web company that happened to sell books instead of thinking of him as a bookseller that wanted to be online.
Can't speak directly for Chuck, but probably not. Does the company see the software they build as a living organism that needs care and maintenance, as well as adaptation to new technologies and platforms, and do they expect to make more software to solve other business problems in the future?
Hard to tell when hiring one or two engineers what the company's longer-term intentions are.
Tech companies have employees on staff who develop web based applications to transact business with customers.
Non Tech companies have employees on staff who specify a set of requirements for a web tool that will allow the business to do their work and then hire a contractor to build it.
It defines what management sees as being 'core' to their idea of what value the business is creating.