The more coherent litmus test for "technology company" is the role of R&D within the organization, irrespective of line-of-business.
On the clearly non-tech company side are those which purely consume off-the-shelf products. Your neighborhood bodega probably does most of its business through computerized cash register and credit card terminal, but is unambiguously not a technology company.
Then there's a vast middle ground where the company engages consultants and maybe a small in-house IT cost center to integrate and operate off-the-shelf products. Some of the more sophisticated players have unique enough requirements that they start to commission their own technology (or at least customizations to their existing SAP, Oracle, etc). This could be done internally or through contractors, but not as a core competency. Emphasis is on meeting requirements as specified by the business and at the lowest cost. Most banks would fit in this category. People who operate the line of business run the company; IT are support staff who occasionally automate the even-lower-status support staff.
Then there are companies whose core competency is R&D, whose technology teams proactively develop and augment products on their own initiative, whose leadership views R&D as an investment/competitive advantage/core of the business rather than as a cost of doing business. This tends to reflect in the social status and compensation of engineers. Engineers (and product and design engineering management) run the company; everyone else is support staff for whatever drudgery engineering hasn't automated yet. This is what we say when we mean "technology company."
But the only way to be a pure technology company is to license your tech to others to commercialize, and even then, by your logic, that would make you a patent licensing company. Most tech companies have a line of business. It could be advertising, entertainment, retail, transportation, trading... anything under the sun. The difference is how they go about it.