Programmatic Testing: A style of software testing where tests are executed without human interaction.
When I say "this test can be automated", I don't mean a computer can magically replace a sapient process to create a useful test. All test design and methodology is devised using a sapient processes (obviously). However, testcase generation, data generation, test execution, results analysis, etc, can often be done without human interaction. This has been described using terms like: "Automated Testing, "Test Automation", etc. Among some of the more vocal people in the testing community, these terms are considered confusing, inaccurate, harmful, or just a cover up for what is actually a non-scripted test
Automated and Manual Testing are not mutually exclusive. Rather, they are at either end of a continuum. A test isn't "automated" or "manual". It falls somewhere on that continuum. Each step of the process has some mixture of the elements.
I have also heard the term: "Computer Assisted Testing".
I don't like this term because it tries to place a name on the entire testing process. I am not referring to the entire process, so this becomes confusing talk to me. I am referring to a style of test execution.
Therefore, to clear up ambiguity of terminology, I now refer to "Automation" as "Programmatic Testing". This refers to a style of test execution where tests are executed in a non-interactive manner.
I you have a tool, a test framework, a software library, a program/script, functions for checking conditions, or a mechanism for reporting errors and stats, then you are doing a form of programmatic testing.