Unit testing - as I understand it - testing individual units of code. It is relatively low level and is usually developed at the same time as the code itself.
To do this, you also need to work in the code, and ultimately the code that performs these tests is a testing tool, even if for some reason you are not using the framework.
So, no, if you are not using testing tools or testing code, you are not doing unit testing.
Theoretically, you can do integration testing manually, but it is still unreliable because people tend to be inconsistent and expensive because people are slower than cars.
Ultimately, the more tests you can automate, the faster and more accurate your tests will be, and the more you free up your QA character to test things that can only be verified manually.
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