It is absolutely not a computer, though. They observed the observable astrological phenomena was cyclical and made a system of gears for their periods to predict it's state. It's interesting, but for context it is centuries younger than the first Greek geared clocks, so it's definitely not something we should be too surprised that existed.
It is absolutely a special–purpose computer. Of course it’s not a general–purpose programmable computer such as the ones we use today, but it is still a computer.
What is your dictionary definition of computer then? Is a sundial a computer because it tells the time? Is an abacus, or a Genaille–Lucas ruler, a computer?
A special–purpose computer is any device which computes an answer to a specific question given some input or initial conditions. An abacus or slide rule is not a computer because they are not designed for a specific question or problem (but they can of course be an aid to computation).
With the Antikythera mechanism, the operator sets the machine to the current date, and then turns the crank to advance the date. The dials on the face would continuously show the positions of several planets, the phase of the moon, the eclipses, and possibly one or two other things.
Charles Babbage’s Difference Engine computed the values of polynomials. The operator would set the coefficients of the polynomial, then turn the crank to compute successive values of that polynomial for successive values of X. He never finished it though.
The firing computer on a WWII destroyer computed the correct elevation of the guns, based on inputs that encoded the position and velocity of the ship, the position and velocity of the target, and several other variables. https://youtu.be/gwf5mAlI7Ug?t=45
General–purpose computers take as input a _program_, and the program specifies the problem to be solved. You could load a program for computing eclipses one minute, and the next minute load a program for computing polynomials or firing solutions, or load one to check your email. The hardware hasn’t changed, only the program and inputs have changed.
Contrast that with the Difference Engine: to make the difference engine compute eclipses or firing solutions, you would have to swap out all of the gears and shafts, and build a different frame to hold them all. It would be turned into a completely different machine.
I had a similar thought and I noticed it wasn’t referred to in the video as a computer, only the title. I’d describe it as a model although I think it fits the common definition of a computer too.
