There is a distinction between computer and stored program computers, and you're right that the first commercially available stored program computer appeared in 1951. (Though, interestingly, there was a 1936 patent application on the idea in Germany. And a barely functioning one was actually built in 1941.)

But Turing complete computers predated that. In fact Turing's own design for a Turing machine was NOT a stored program computer. And you're right that the modern idea of programming postdated stored program computers.

But computers are older. As you pointed out, arguably thousands of years older.

However I maintain that automatic tabulating machines were on the path to modern computers. Early accounting applications were based on them, as were key parts of the technology used. Like punch cards.

I agree that "automatic tabulating machines were on the path to modern computers" were on the path to modern computers. But that does not in any way negate that there was an inflection point in the 1950s and leading into the 1960s that represents the birth of what we mean by the word computer (in English). The machines that I learned programming on in the 1960s were qualitatively different from those things "on the path" to them, but are recognizably the same species of machine as what we work on today - which is to say, they were Von Neumann architecture, Turing complete, off-the-shelf, generally available computers for which one wrote code - not calculators, tabulating machines, or hardware reconfigurable logic engines, and not laboratory experiments aiming toward that. The only thing missing from them that is inherent in our contemporary approach to information is the notion of ubiquitous networking of devices to create a broad and broadly accessible information and calculation landscape. That was still a laboratory project in 1970.

Does that matter? Depends on your point of view. What I meant by "how new it all seemed" then tells the tale here. Baruch could maybe see the future in 1945 because of his experience and vision. But by 1970 or thereabouts, almost anyone who touched it could see it, and millions were increasingly able to touch it, precisely because it was was commercially produced and generally available. Still not ubiquitous, as they are today, computers were nevertheless showing up in every corner of life.

So it's not that I think computers or information as a processing as a concept, or even as an occasional reality, were new around 1970. But computers as a phenomenon that would define the way interact with each other and the world, and computer programming as a skill that anyone could acquire, really were.