Putting it in simplest way (as per my thinking), test automation is writing a software (programs) to test another software.
With this said, I'm a little confused about a few of the statements in your question description.
I think automated testing is when you do nothing about tests. Just
giving the program to a test generator and it makes some inputs and
What exactly do you mean when you say you do nothing about tests?
Who writes the tests?
How are the tests invoked and executed? Who wrote that process?
What is this test generator that you give your program to?
How does it decide what inputs to make and whether the result is right or wrong?
Also obviously, oracle in unit testing with tools is a human. So it is
manual, because a human determines the expected output results.
Whether you write programs to execute your tests or execute them yourself without the help of programs/tools, it is always you who determines the EXPECTED output. Programs/Tools don't have brains of their own to think and decide.
So in real sense there is nothing called automated or manual testing.
It's simply testing. Yes, sometimes to help with quicker executions or for any other reason, you may use tools and programs to aide your testing. It's like you may get bored repeating same steps over and over. So you would think of writing a program to perform those repetitions for you. Still it's always you in control of what can be programmed and how that program will execute and list out the result. It's always you who decides what input is to be given to the program and which results will be considered and right or wrong behavior of the software that you are testing.
Say you may also program the process of invoking your testing programs and generating and sending out reports to concerned people. People do that in CI/Cd pipelines usually.
But, at the end of the day it's all you (a human) who did that. Nothing happened on it own.
So don't fall into the pit of what's manual and what's automated. Focus on testing and wherever you think use or tools/programs can enhance your work, go ahead and do it. But, never forget, it's all just TESTING.